IRLF 


THE   THEORY   OF   MEASUREMENTS 


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THE  THEORY  OF 

MEASUREMENTS 


BY 

A.  DE  FOREST   PALMER,   PH.D. 

Associate  Professor  of  Physics  in  Brown  University. 


McGRAW-HILL   BOOK   COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 
6  BOUVERIE  STREET,  LONDON,  E.G. 

1912 


COPYRIGHT,  1912, 

BY   THE 

McGRAW-HILL  BOOK  COMPANY 


Stanbopc  jjbress 

H.GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE. 


THE  function  of  laboratory  instruction  in  physics  is  twofold. 
Elementary  courses  are  intended  to  develop  the  power  of  discrimi- 
nating observation  and  to  put  the  student  in  personal  contact  with 
the  phenomena  and  general  principles  discussed  in  textbooks  and 
lecture  demonstrations.  The  apparatus  provided  should  be  of  the 
simplest  possible  nature,  the  experiments  assigned  should  be  for 
the  most  part  qualitative  or  only  roughly  quantitative,  and  emphasis 
should  be  placed  on  the  principles  illustrated  rather  than  on  the 
accuracy  of  the  necessary  measurements.  On  the  other  hand, 
laboratory  courses  designed  for  more  mature  students,  who  are 
supposed  to  have  a  working  knowledge  of  fundamental  principles, 
are  intended  to  give  instruction  in  the  theory  and  practice  of  the 
methods  of  precise  measurement  that  underlie  all  effective  research 
and  supply  the  data  on  which  practical  engineering  enterprises  are 
based.  They  should  also  develop  the  power  of  logical  argument 
and  expression,  and  lead  the  student  to  draw  rational  conclusions 
from  his  observations.  The  instruments  provided  should  be  of 
standard  design  and  efficiency  in  order  that  the  student  may  gain 
practice  in  making  adjustments  and  observations  under  as  nearly 
as  may  be  the  same  conditions  that  prevail  in  original  investigation. 

Measurements  are  of  little  value  in  either  research  or  engineering 
applications  unless  the  precision  with  which  they  represent  the 
measured  magnitude  is  definitely  known.  Consequently,  the  stu- 
dent should  be  taught  to  plan  and  execute  proposed  measurements 
within  definitely  prescribed  limits  and  to  determine  the  accuracy 
of  the  results  actually  attained.  Since  the  treatment  of  these 
matters  in  available  laboratory  manuals  is  fragmentary  and  often 
very  inadequate  if  not  misleading,  the  author  some  years  ago  under- 
took to  impart  the  necessary  instruction,  in  the  form  of  lectures, 
to  a  class  of  junior  engineering  students.  Subsequently,  textbooks 
on  the  Theory  of  Errors  and  the  Method  of  Least  Squares  were 
adopted  but  most  of  the  applications  to  actual  practice  were  still 
given  by  lecture.  The  present  treatise  is  the  result  of  the  experi- 


257860 


VI  PREFACE 

ence  gained  with  a  number  of  succeeding  classes.  It  has  been 
prepared  primarily  to  meet  the  needs  of  students  in  engineering 
and  advanced  physics  who  have  a  working  knowledge  of  the  differ- 
ential and  integral  calculus.  It  is  not  intended  to  supersede  but 
to  supplement  the  manuals  and  instruction  sheets  usually  employed 
in  physical  laboratories,  Consequently,  particular  instruments  and 
methods  of  measurement  have  been  described  only  in  so  far  as  they 
serve  to  illustrate  the  principles  under  discussion. 

The  usefulness  of  such  a  treatise  was  suggested  by  the  marked 
tendency  of  laboratory  students  to  carry  out  prescribed  work  in  a 
purely  automatic  manner  with  slight  regard  for  the  significance  or 
the  precision  of  their  measurements.  Consequently,  an  endeavor 
has  been  made  to  develop  the  general  theory  of  measurements  and 
the  errors  to  which  they  are  subject  in  a  form  so  clear  and  concise 
that  it  can  be  comprehended  and  applied  by  the  average  student 
with  the  prescribed  previous  training.  To  this  end,  numerical  ex- 
amples have  been  introduced  and  completely  worked  out  whenever 
this  course  seemed  likely  to  aid  the  student  in  obtaining  a  thorough 
grasp  of  the  principles  they  illustrate.  On  the  other  hand,  inherent 
difficulties  have  not  been  evaded  and  it  is  not  expected,  or  even 
desired,  that  the  student  will  be  able  to  master  the  subject  without 
vigorous  mental  effort. 

The  first  seven  chapters  deal  with  the  general  principles  that 
underlie  all  measurements,  with  the  nature  and  distribution  of  the 
errors  to  which  they  are  subject,  and  with  the  methods  by  which 
the  most  probable  result  is  derived  from  a  series  of  discordant 
measurements.  The  various  types  of  measurement  met  with  in 
practice  are  classified,  and  general  methods  of  dealing  with  each 
of  them  are  briefly  discussed.  Constant  errors  and  mistakes  are 
treated  at  some  length,  and  then  the  unavoidable  accidental  errors 
of  observation  are  explicitly  defined.  The  residuals  corresponding 
to  actual  measurements  are  shown  to  approach  the  true  accidental 
errors  as  limits  when  the  number  of  observations  is  indefinitely 
increased  and  their  normal  distribution  in  regard  to  sign  and  mag- 
nitude is  explained  and  illustrated.  After  a  preliminary  notion  of 
its  significance  has  been  thus  imparted,  the  law  of  accidental  errors 
is  stated  empirically  in  a  form  that  gives  explicit  representation  to 
all  of  the  factors  involved.  It  is  then  proved  to  be  in  conformity 
with  the  axioms  of  accidental  errors,  the  principle  of  the  arithmetical 
ij  and  the  results  of  experience.  The  various  characteristic 


PREFACE  vii 

errors  that  are  commonly  used  as  a  measure  of  the  accidental  errors 
of  given  series  of  measurements  are  clearly  denned  and  their  signifi- 
cance is  very  carefully  explained  in  order  that  they  may  be  used 
intelligently.  Practical  methods  for  computing  them  are  developed 
and  illustrated  by  numerical  examples. 

Chapters  eight  to  twelve  inclusive  are  devoted  to  a  general  dis- 
cussion of  the  precision  of  measurements  based  on  the  principles 
established  in  the  preceding  chapters.  The  criteria  of  accidental 
errors  and  suitable  methods  for  dealing  with  constant  and  systematic 
errors  are  developed  in  detail.  The  precision  measure,  of  the  result 
computed  from  given  observations,  is  defined  and  its  significance  is 
explained  with  the  aid  of  numerical  illustrations.  The  proper  basis 
for  the  criticism  of  reported  measurements  and  the  selection  of 
suitable  numerical  values  from  tables  of  physical  constants  or  other 
published  data  is  outlined ;  and  the  importance  of  a  careful  estimate 
of  the  precision  of  the  data  adopted  in  engineering  and  scientific 
practice  is  emphasized.  The  applications  of  the  theory  of  errors  to 
the  determination  of  suitable  methods  for  the  execution  of  proposed 
measurements  are  discussed  at  some  length  and  illustrated. 

In  chapter  thirteen,  the  relation  between  measurement  and  re- 
search is  pointed  out  and  the  general  methods  of  physical  research 
are  outlined.  Graphical  methods  of  reduction  and  representation 
are  explained  and  some  applications  of  the  method  of  least  squares 
are  developed.  The  importance  of  timely  and  adequate  publication, 
or  other  report,  of  completed  investigations  is  emphasized  and  some 
suggestions  relative  to  the  form  of  such  reports  are  given 

Throughout  the  book,  particular  attention  is  paid  to  methods  of 
computation  and  to  the  proper  use  of  significant  figures.  For  the 
convenience  of  the  student,  a  number  of  useful  tables  are  brought 
together  at  the  end  of  the  volume. 

A.  DE  FOREST  PALMER. 

BROWN  UNIVERSITY, 
July,  1912. 


CONTENTS. 


PAGE 

PREFACE v 

CHAPTER  I. 

GENERAL  PRINCIPLES 1 

Introduction  —  Measurement  and  Units  —  Fundamental  and 
Derived  Units  —  Dimensions  of  Units  —  Systems  of  Units  in  Gen- 
eral Use  —  Transformation  of  Units. 


CHAPTER  II. 

MEASUREMENTS 11 

Direct  Measurements  —  Indirect  Measurements  —  Classification  of 
Indirect  Measurements  —  Determination  of  Functional  Relations 
—  Adjustment,  Setting,  and  Observation  of  Instruments  —  Record 
of  Observations  —  Independent,  Dependent,  and  Conditioned 
Measurements  —  Errors  and  the  Precision  of  Measurements  —  Use 
of  Significant  Figures  —  Adjustment  of  Measurements  —  Discus- 
sion of  Instruments  and  Methods. 


CHAPTER  III. 

CLASSIFICATION  OF  ERRORS 23 

Constant  Errors  —  Personal  Errors  —  Mistakes  —  Accidental 
Errors  —  Residuals  —  Principles  of  Probability. 

CHAPTER  IV. 

THE  LAW  OF  ACCIDENTAL  ERRORS 29 

Fundamental  Propositions  —  Distribution  of  Residuals  —  Proba- 
bility of  Residuals  —  The  Unit  Error  —  The  Probability  Curve  — 
Systems  of  Errors  —  The  Probability  Function  —  The  Precision 
Constant  —  Discussion  of  the  Probability  Function  —  The  Proba- 
bility Integral  —  Comparison  of  Theory  and  Experience  —  The 
Arithmetical  Mean. 

CHAPTER   V. 

CHARACTERISTIC  ERRORS 44 

The  Average  Error  —  The  Mean  Error  —  The  Probable  Error  — 
Relations  between  the  Characteristic  Errors  —  Characteristic 
Errors  of  the  Arithmetical  Mean  —  Practical  Computation  of 
Characteristic  Errors  —  Numerical  Example  —  Rules  for  the  Use 
of  Significant  Figures. 

CHAPTER  VI. 

MEASUREMENTS  OF  UNEQUAL  PRECISION 61 

Weights  of  Measurements  —  The  General  Mean  —  Probable  Error 
of  the  General  Mean  —  Numerical  Example. 

ix 


x  CONTENTS 

CHAPTER  VII. 

PAGE 

THE  METHOD  OF  LEAST  SQUARES 72 

Fundamental  Principles  —  Observation  Equations  —  Normal  Equa- 
tions —  Solution  with  Two  Independent  Variables  —  Adjustment  of 
the  Angles  about  a  Point  —  Computation  Checks  —  Gauss's  Method 
of  Solution  —  Numerical  Illustration  of  Gauss's  Method  —  Con- 
ditioned Quantities. 

CHAPTER  VIII. 

PROPAGATION  OP  ERRORS 95 

Derived  Quantities  —  Errors  of  the  Function  Xi  ±  Xz  ±  X3  ± 
.  .  .  ±  Xq  —  Errors  of  the  Function  ai-Xi  ±  0:2^2  ±  013X3  =h  .  .  . 
±  aqXq —  Errors  of  the  Function  F  (Xi,  X?,  .  .  .  ,  Xq)  —  Example 
Introducing  the  Fractional  Error  —  Fractional  Error  of  the  Func- 
tion aX!±n>  X  X2±n'  X  ...  X  Xq±n*. 

CHAPTER  IX. 

ERRORS  OF  ADJUSTED  MEASUREMENTS 105 

Weights  of  Adjusted  Measurements  —  Probable  Error  of  a  Single 
Observation  —  Application  to  Problems  Involving  Two  Unknowns 

—  Application  to  Problems  Involving  Three  Unknowns. 

CHAPTER  X. 

DISCUSSION  OF  COMPLETED  OBSERVATIONS 117 

Removal  of  Constant  Errors  — •  Criteria  of  Accidental  Errors  — 
Probability  of  Large  Residuals  —  Chauvenet's  Criterion  —  Preci- 
sion of  Direct  Measurements — Precision  of  Derived  Measurements 

—  Numerical  Example. 

CHAPTER  XI. 

DISCUSSION  OF  PROPOSED  MEASUREMENTS 144 

Preliminary  Considerations  —  The  General  Problem  —  The  Pri- 
mary Condition  —  The  Principle  of  Equal  Effects  —  Adjusted  Effects 

—  Negligible  Effects  —  Treatment  of  Special  Functions  —  Numerical 
Example. 

CHAPTER  XII. 

BEST  MAGNITUDES  FOR  COMPONENTS 165 

Statement  of  the  Problem  —  General  Solutions  —  Special  Cases  — 
Practical  Examples  —  Sensitiveness  of  Methods  and  Instruments. 

CHAPTER  XIII. 

RESEARCH 192 

Fundamental  Principles  —  General  Methods  of  Physical  Research 

—  Graphical  Methods  of  Reduction  —  Application  of  the  Method 
of  Least  Squares  —  Publication. 

TABLES 212 

INDEX..  245 


LIST   OF   TABLES. 


PAGE 

I.  DIMENSIONS  OF  UNITS 212 

II.  CONVERSION  FACTORS 213 

III.  TRIGONOMETRICAL  RELATIONS 215 

IV.  SERIES 217 

V.  DERIVATIVES 219 

VI.  SOLUTION  OF  EQUATIONS 220 

VII.  APPROXIMATE  FORMULA 221 

VIII.  NUMERICAL  CONSTANTS 222 

IX.  EXPONENTIAL  FUNCTIONS  ex  AND  e~x 223 

X.  EXPONENTIAL  FUNCTIONS  e*2  AND  e~xZ 224 

XI.  THE  PROBABILITY  INTEGRAL  PA 225 

XII.  THE  PROBABILITY  INTEGRAL  Ps 226 

XIII.  CHAUVENET'S  CRITERION 226 

XIV.  FOR  COMPUTING  PROBABLE  ERRORS  BY  FORMULAE  (31)  AND  (32).  227 
XV.  FOR  COMPUTING  PROBABLE  ERRORS  BY  FORMULA  (34) 228 

XVI.  SQUARES  OF  NUMBERS 229 

XVII.  LOGARITHMS;  1000  TO  1409 231 

XVIII.  LOGARITHMS 232 

XIX.  NATURAL  SINES 234 

XX.  NATURAL  COSINES 236 

XXI.  NATURAL  TANGENTS 238 

XXII.  NATURAL  COTANGENTS 240 

XXIII.  RADIAN  MEASURE.  .  242 


THE 
THEOEY   OF   MEASUREMENTS 


CHAPTER  I. 
GENERAL  PRINCIPLES. 

i.  Introduction.  —  Direct  observation  of  the  relative  position 
and  motion  of  surrounding  objects  and  of  their  similarities  and 
differences  is  the  first  step  in  the  acquisition  of  knowledge. 
Such  observations  are  possible  only  through  the  sensations  pro- 
duced by  our  environment,  and  the  value  of  the  knowledge  thus 
acquired  is  dependent  on  the  exactness  with  which  we  corre- 
late these  sensations.  Such  correlation  involves  a  quantitative 
estimate  of  the  relative  intensity  of  different  sensations  and  of 
their  time  and  space  relations.  As  our  estimates  become  more 
and  more  exact  through  experience,  our  ideas  regarding  the 
objective  world  are ,  gradually  modified  until  they  represent 
the  actual  condition  of  things  with  a  considerable  degree  of 
precision. 

The  growth  of  science  is  analogous  to  the  growth  of  ideas. 
Its  function  is  to  arrange  a  mass  of  apparently  isolated  and  un- 
related phenomena  in  systematic  order  and  to  determine  the  in- 
terrelations between  them.  For  this  purpose,  each  quantity  that 
enters  into  the  several  phenomena  must  be  quantitatively  deter- 
mined, while  all  other  quantities  are  kept  constant  or  allowed 
to  vary  by  a  measured  amount.  The  exactness  of  the  relations 
thus  determined  increases  with'  the  precision  of  the  measure- 
ments and  with  the  success  attained  in  isolating  the  particular 
phenomena  investigated. 

A  general  statement,  or  a  mathematical  formula,  that  ex- 
presses the  observed  quantitative  relation  between  the  different 
magnitudes  involved  in  any  phenomenon  is  called  the  law  of 
that  phenomenon.  As  here  used,  the  word  law  does  not  mean 

1 


2  THE  THEORY  OF  MEASUREMENTS          [ART.  2 

that  the  phenomenon  must  follow  the  prescribed  course,  but 
that,  under  the  given  conditions  and  within  the  limits  of  error 
and  the  range  of  our  measurements,  it  has  never  been  found  to 
deviate  from  that  course.  In  other  words,  the  laws  of  science 
are  concise  statements  of  our  present  knowledge  regarding 
phenomena  and  their  relations.  As  we  increase  the  range  and 
accuracy  of  our  measurements  and  learn  to  control  the  condi- 
tions of  experiment  more  definitely,  the  laws  that  express  our 
results  become  more  exact  and  cover  a  wider  range  of  phenomena. 
Ultimately  we  arrive  at  broad  generalizations  from  which  the 
laws  of  individual  phenomena  are  deducible  as  special  cases. 

The  two  greatest  factors  in  the  progress  of  science  are  the 
trained  imagination  of  the  investigator  and  the  genius  of 
measurement.  To  the  former  we  owe  the  rational  hypotheses 
that  have  pointed  the  way  of  advance  and  to  the  latter  the 
methods  of  observation  and  measurement  by  which  the  laws  of 
science  have  been  developed. 

2.  Measurement  and  Units.  —  To  measure  a  quantity  is  to 
determine  the  ratio  of  its  magnitude  to  that  of  another  quan- 
tity, of  the  same  kind,  taken  as  a  unit.  The  number  that 
expresses  this  ratio  may  be  either  integral  or  fractional  and  is 
called  the  numeric  of  the  given  quantity  in  terms  of  the  chosen 
unit.  In  general,  if  Q  represents  the  magnitude  of  a  quantity, 
U  the  magnitude  of  the  chosen  unit,  and  N  the  corresponding 
numeric  we  have 

Q  =  NU,  (I) 

which  is  the  fundamental  equation  of  measurement.  The  two 
factors  N  and  U  are  both  essential  for  the  exact  specification  of 
the  magnitude  Q.  For  example:  the  length  of  a  certain  line 
is  five  inches,  i.e.,  the  line  is  five  times  as  long  as  one  inch.  It 
is  not  sufficient  to  say  that  the  length  of  the  line  is  five;  for  in 
that  case  we  are  uncertain  whether  its  length  is  five  inches,  five 
feet,  or  five  times  some  other  unit. 

Obviously,  the  absolute  magnitude  of  a  quantity  is  independent 
of  the  units  with  which  we  choose  to  measure  it.  Hence,  if  we 
adopt  a  different  unit  U',  we  shall  find  a  different  numeric  N' 

such  that 

Q  =  N'U',  (II) 

and  consequently 

NU  =  N'U', 


ART.  2]  •'    GENERAL  PRINCIPLES  3 

or  $-^-  (HI) 

Equation  (III)  expresses  the  general  principle  involved  in  the 
transformation  of  units  and  shows  that  the  numeric  varies  in- 
versely as  the  magnitude  of  the  unit;  i.e.,  if  U  is  twice  as  large 
as  U',  N  will  be  only  one-half  as  large  as  N'.  To  take  a  con- 
crete example:  a  length  equal  to  ten  inches  is  also  equal  to 
25.4  centimeters  approximately.  In  this  case  N  equals  ten, 
N'  equals  25.4,  U  equals  one  inch,  and  Ur  equals  one  centi- 

Nf 
meter.     The  ratio  of  the  numerics  -^  is  2.54  and  hence  the 

inverse  ratio  of  the  units      -,  is  also  2.54,  i.e.,  one  inch  is  equal  to 


2.54  centimeters. 

Equation  (III)  may  also  be  written  in  the  form 


(IV) 


which  shows  that  the  numeric  of  a  given  quantity  relative  to  the 
unit  U  is  equal  to  its  numeric  relative  to  the  unit  U'  multiplied 

w 

by  the  ratio  of  the  unit  Uf  to  the  unit  U.     The  ratio  -jj  is  called 

the  conversion  factor  for  the  unit  Uf  in  terms  of  the  unit  U. 
It  is  equal  to  the  number  of  units  U  in  one  unit  U',  and  when 
multiplied  by  the  numeric  of  a  quantity  in  terms  of  U'  gives 
the  numeric  of  the  same  quantity  in  terms  of  U.  The  con- 
version factor  for  transformation  in  the  opposite  direction,  i.e., 

from  U  to  U',  is  obviously  the  inverse  of  the  above,  or  -==  •     In 

general,  the  numerator  of  the  conversion  factor  is  the  unit  in 
which  the  magnitude  is  already  expressed  and  the  denominator 
is  the  unit  to  which  it  is  to  be  transformed.  For  example: 
one  inch  is  approximately  equal  to  2.54  centimeters,  hence  the 
numeric  of  a  length  in  centimeters  is  about  2.54  times  its  numeric 
in  inches.  Conversely,  the  numeric  in  inches  is  equal  to  the 
numeric  in  centimeters  divided  by  2.54  or  multiplied  by  the 
reciprocal  of  this  number. 

In  so  far  as  the  theory  of  mensuration  and  the  attainable 
accuracy  of  the  result  are  concerned,  measurements  may  be  made 
in  terms  of  any  arbitrary  unite  and,  in  fact,  the  adoption  oisuch 


4  THE  THEORY  OF  MEASUREMENTS         [ART.  3 

units  is  frequently  convenient  when  we  are  concerned  only  with 
relative  determinations.  In  general,  however,  measurements  are 
of  little  value  unless  they  are  expressed  in  terms  of  generally 
accepted  units  whose  magnitude  is  accurately  known.  Some 
such  units  have  come  into  use  through  common  consent  but  most 
of  them  have  been  fixed  by  government  enactment  and  their  per- 
manence is  assured  by  legal  standards  whose  relative  magnitudes 
have  been  accurately  determined.  Such  primary  standards,  pre- 
served by  various  governments,  have,  in  many  cases,  been  very 
carefully  intercompared  and  their  conversion  factors  are  accu- 
rately known.  Copies  of  the  more  important  primary  standards 
may  be  found  in  all  well-equipped  laboratories  where  they  are 
preserved  as  the  secondary  standards  to  which  all  exact  measure- 
ments are  referred.  Carefully  made  copies  are,  usually,  sufficiently 
accurate  for  ordinary  purposes,  but,  when  the  greatest  precision 
is  sought,  their  exact  magnitude  must  be  determined  by  direct 
comparison  with  the  primary  standards.  The  National  Bureau 
of  Standards  at  Washington  makes  such  comparisons  and  issues 
certificates  showing  the  errors  of  the  standards  submitted  for 
test. 

3.  Fundamental  and  Derived  Units.  —  Since  the  unit  is,  neces- 
sarily, a  quantity  of  the  same  kind  as  the  quantity  measured,  we 
must  have  as  many  different  units  as  there  are  different  kinds  of 
quantities  to  be  measured.  Each  of  these  units  might  be  fixed 
by  an  independent  arbitrary  standard,  but,  since  most  measur- 
able quantities  are  connected  by  definite  physical  relations,  it  is 
more  convenient  to  define  our  units  in  accordance  with  these 
relations.  Thus,  measured  in  terms  of  any  arbitrary  unit,  a 
uniform  velocity  is  proportional  to  the  distance  described  in 
unit  time;  but,  if  we  adopt  as  our  unit  such  a  velocity  that  the 
unit  of  length  is  traversed  in  the  unit  of  time,  the  factor  of  pro- 
portionality is  unity  and  the  velocity  is  equal  to  the  ratio  of  the 
space  traveled  to  the  elapsed  time. 

Three  independently  defined  units  are  sufficient,  in  connection 
with  known  physical  relations,  to  fix  the  value  of  most  of  the 
other  units  used  in  physical  measurements.  We  are  thus  led  to 
distinguish  two  classes  of  units;  the  three  fundamental  units, 
defined  by  independent  arbitrary  standards,  and  the  derived 
units,  fixed  by  definite  relations  between  the  fundamental  units. 
The  .magnitude,  and  to  some  extent  the  choice,  of  the  fundamental 


ART.  4]  GENERAL  PRINCIPLES  5 

units  is  arbitrary,  but  when  definite  standards  for  each  of  these 
units  have  been  adopted  the  magnitude  of  all  of  the  derived  units 
is  fixed. 

For  convenience  in  practice,  legal  standards  have  been  adopted 
to  represent  some  of  the  derived  units.  The  precision  of  these 
standards  is  determined  by  indirect  comparison  with  the  standards 
representing  the  three  fundamental  units.  Such  comparisons  are 
based  on  the  known  relations  between  the  fundamental  and  de- 
rived units  and  are  called  absolute  measurements.  The  practical 
advantage  gained  by  the  use  of  derived  standards  lies  in  the  fact 
that  absolute  measurements  are  generally  very  difficult  and  require 
great  skill  and  experience  in  order  to  secure  a  reasonable  degree 
of  accuracy.  On  the  other  hand,  direct  comparison  of  derived 
quantities  of  the  same  kind  is  often  a  comparatively  simple 
matter  and  can  be  carried  out  with  great  precision. 

4.  Dimensions  of  Units.  —  The  dimensions  of  a  unit  is  a 
mathematical  formula  that  shows  how  its  magnitude  is  related 
to  that  of  the  three  fundamental  units.  In  writing  such  formulae, 
the  variables  are  usually  represented  by  capital  letters  inclosed 
in  square  brackets.  Thus,  [M],  [L]  and  [T]-  represent  the  dimen- 
sions of  the  units  of  mass,  length  and  time  respectively. 

Dimensional  formulae  and  ordinary  algebraic  equations  are 
essentially  different  in  significance.  The  former  shows  the  rela- 
tive variation  of  units,  while  the  latter  expresses  a  definite  mathe- 
matical relation  between  the  numerics  of  measurable  quantities. 
Thus  if  a  point  in  uniform  motion  describes  the  distance  L  in  the 
time  T  its  velocity  V  is  defined  by  the  relation 

V  =  Y  (V) 

Since  L  and  T  are  concrete  quantities  of  different  kind,  the  right- 
hand  member  of  this  equation  is  not  a  ratio  in  the  strict  arithmet- 
ical sense;  i.e.,  it  cannot  be  represented  by  a  simple  abstract  num- 
ber. Hence,  in  virtue  of  the  definite  physical  relation  expressed 
by  equation  (V),  we  are  led  to  extend  our  idea  of  ratio  to  include 
the  case  of  concrete  quantities.  From  this  point  of  view,  the  ratio 
of  two  quantities  expresses  the  rate  of  change  of  the  first  quantity 
with  respect  to  the  second.  It  is  a  concrete  quantity  of  the  same 
kind  as  the  quantity  it  serves,  to  define.  As  an  illustration,  con- 
sider the  meaning  of  equation  (V).  Expressed  in  words,  it  is  "  the 


6  THE  THEORY  OF  MEASUREMENTS         [ART.  4 

velocity  of  a  point,  in  uniform  motion,  is  equal  to  the  time  rate  at 
which  it  moves  through  space." 

If  we  represent  the  units  of  velocity,  length,  and  time  by  [7], 
[L],  and  [T\,  respectively,  and  the  corresponding  numerics  by  v, 
I,  and  t,  we  have  by  equation  (I),  article  two, 

F  =  v(V],        L  =  l(L],         T  =  t[T], 
and  equation  (V)  becomes 

w-m-i' 

or 


[V][T]   t 

Since,  by  definition,  [V]  and  |~l  are  quantities  of  the  same  kind, 

their  ratio  can  be  expressed  by  an  abstract  number  k  and  equation 
(VI)  may  be  written  in  the  form 

v  =  kl,  (VII) 

which  is  an  exact  numerical  equation  containing  no  concrete 
quantities. 

The  numerical  value  of  the  constant  k  obviously  depends  on 
the  units  with  which  L,  T,  and  V  are  measured.  If  we  define  the 
unit  of  velocity  by  the  relation 

ryi-M 
[TV 
or,  as  it  is  more  often  written, 

[F]  =  [L!T-']f  (VIII) 

k  becomes  equal  to  unity  and  the  relation  (VII)  between  the 
numerics  of  velocity,  length,  and  time  reduces  to  the  simple  form 


The  foregoing  argument  illustrates  the  advantage  to  be  gained 
by  defining  derived  units  in  accordance  with  the  physical  rela- 
tions on  which  they  depend.  By  this  means  we  eliminate  the 
often  incommensurable  constants  of  proportionality  such  as  k 
would  be  if  the  unit  of  velocity  were  defined  in  any  other  way 
than  by  equation  (VIII). 


ART.  5]  GENERAL  PRINCIPLES  7 

The  expression  on  the  right-hand  side  of  equation  (VIII)  is  the 
dimensions  of  the  unit  of  velocity  when  the  units  of  length,  mass, 
and  time  are  chosen  as  fundamental.  The  dimensions  of  any 
other  units  may  be  obtained  by  the  method  outlined  above  when 
we  know  the  physical  relations  on  which  they  depend.  The  form 
of  the  dimensional  formula  depends  on  the  units  we  choose  as 
fundamental,  but  the  general  method  of  derivation  is  the  same  in 
all  cases.  As  an  exercise  to  fix  these  ideas  the  student  should 
verify  the  following  dimensional  formulae:  choosing  [M],  [L],  and 
[T]  as  fundamental  units,  the  dimensions  of  the  units  of  area, 
acceleration,  and  force  are  [L2],  [LT~2],  and  [MLT~2]  respectively. 
As  an  illustration  of  the  effect  of  a  different  choice  of  fundamental 
units,  it  may  be  shown  that  the  dimensions  of  the  unit  of  mass  is 
[FL^T2]  when  the  units  of  length  [L],  force  [F],  and  time  [T]  are 
chosen  as  fundamental.  The  dimensions  of  some  important 
derived  units  are  given  in  Table  I  at  the  end  of  this  volume. 

5.  Systems  of  Units  in  General  Use.  —  Consistent  systems 
of  units  may  differ  from  one  another  by  a  difference  in  the  choice 
of  fundamental  units  or  by  a  difference  in  the  magnitude  of  the 
particular  fundamental  units  adopted.  The  systems  in  common 
use  illustrate  both  types  of  difference. 

Among  scientific  men,  the  so-called  c.g.s.  system  is  almost 
universally  adopted,  and  the  results  of  scientific  investigations 
are  seldom  expressed  in  any  other  units.  The  advantage  of  such 
uniformity  of  choice  is  obvious.  It  greatly  facilitates  the  com- 
parison of  the  results  of  different  observers  and  leads  to  general 
advance  in  our  knowledge  of  the  phenomena  studied.  The  units 
of  length,  mass,  and  time  are  chosen  as  fundamental  in  this 
system  and  the  particular  values  assigned  to  them  are  the  centi- 
meter for  the  unit  of  length,  the  gram  for  the  unit  of  mass,  and 
the  mean  solar  second  for  the  unit  of  time. 

The  units  used  commercially  in  England  and  the  United  States 
of  America  are  far  from  systematic,  as  most  of  the  derived  units 
are  arbitrarily  defined.  So  far  as  they  follow  any  order,  they 
form  a  length-mass-time  system  in  which  the  unit  of  length  is  the 
foot,  the  unit  of  mass  is  the  mass  of  a  pound,  and  the  unit  of  time 
is  the  second.  This  system  was  formerly  used  quite  extensively 
by  English  scientists  and  the  results  of  some  classic  investigations 
are  expressed  in  such  units. 

English  and  American  engineers  find  it  more  convenient  to  use 


8  THE  THEORY  OF  MEASUREMENTS         [ART.  6 

a  system  in  which  the  fundamental  units  are  those  of  length, 
force,  and  time.  The  particular  units  chosen  are  the  foot  as  the 
unit  of  length,  the  pound's  weight  at  London  as  the  unit  of  force, 
and  the  mean  solar  second  as  the  unit  of  time.  We  shall  see  that 
this  is  equivalent  to  a  length-mass-time  system  in  which  the  units 
of  length  and  time  are  the  same  as  above  and  the  unit  of  mass  is 
the  mass  of  32.191  pounds. 

6.  Transformation  of  Units.  —  When  the  relative  magnitude 
of  corresponding  fundamental  units  in  two  systems  is  known,  a 
result  expressed  in  one  system  can  be  reduced  to  the  other  with 
the  aid  of  the  dimensions  of  the  derived  units  involved.  Thus: 
let  Ac  represent  the  magnitude  of  a  square  centimeter,  At  the 
magnitude  of  a  square  inch,  Nc  the  numeric  of  a  given  area  when 
measured  in  square  centimeters,  and  Ni  the  numeric  of  the  same 
area  when  measured  in  square  inches;  then,  from  equation  (IV), 
article  two,  we  have 


But  if  Lc  is  the  magnitude  of  a  centimeter  and  LI  that  of  an  inch, 
Ai  is  equal  to  Lf,  and  therefore 


Hence,  the  conversion  factor  -p  for  reducing  square  centimeters 

A-i 

to  square  inches  is  equal  to  the  square  of  the  conversion  factor 

—•  for  reducing  from  centimeters  to  inches.  Now  the  dimensions 
Li 

of  the  unit  of  area  is  [L2],  and  we  see  that  the  conversion  factor 
for  area  may  be  obtained  by  substituting  the  corresponding  con- 
version factor  for  lengths  in  this  dimensional  formula.  This  is  a 
simple  illustration  of  the  general  method  of  transformation  of 
units.  When  the  fundamental  units  in  the  two  systems  differ  in 
magnitude,  but  not  in  kind,  the  conversion  factor  for  correspond- 
ing derived  units  in  the  two  systems  is  obtained  by  replacing  the 
fundamental  units  by  their  respective  conversion  factors  in  the 
dimensions  of  the  derived  units  considered. 

It  should  be  noticed  that  the  fundamental  units  in  the  c.g.s. 
system  are  those  of  length,  mass,  and  time,  while  on  the  engineer's 
system  they  are  length,  force,  and  time.  In  the  latter  system, 


ART.  6]  GENERAL  PRINCIPLES  9 

force  is  supposed  to  be  directly  measured  and  expressed  by  the 
dimensions  [F].  Consequently  the  dimensions  of  the  unit  of 
mass  are  [FL~1T2],  and  the  unit  of  mass  is  a  mass  that  will  acquire 
.  a  velocity  of  one  foot  per  second  in  one  second  when  acted  upon 
by  a  force  of  one  pound's  weight.  For  the  sake  of  definiteness, 
the  unit  of  force  is  taken  as  the  pound's  weight  at  London,  where 
the  acceleration  due  to  gravity  (g)  is  equal  to  32.191  feet  per 
second  per  second.  Otherwise  the  unit  of  force  would  be  variable, 
depending  on  the  place  at  which  the  pound  is  weighed. 

From  Newton's  second  law  of  motion  we  know  that  the  relation 
between  acceleration,  mass,  and  force  is  given  by  the  expression 

/  =  ma. 

For  a  constant  force  the  acceleration  produced  is  inversely  pro- 
portional to  the  mass  moved.  Now  the  mass  of  a  pound  at  London 
is  acted  upon  by  gravity  with  a  force  of  one  pound's  weight,  and,  if 
free,  it  moves  with  an  acceleration  of  32.191  feet  per  second  per 
second.  Hence  a  mass  equal  to  that  of  32.191  pounds  acted 
upon  by  a  force  of  one  pound's  weight  would  move  with  an  acceler- 
ation of  one  foot  per  second  per  second,  i.e.,  it  would  acquire  a 
velocity  of  one  foot  per  second  in  one  second.  Hence  the  unit  of 
mass  in  the  engineer's  system  is  32.191  pounds  mass.  This  unit 
is  sometimes  called  a  slugg,  but  the  name  is  seldom  met  with  since 
engineers  deal  primarily  with  forces  rather  than  masses,  and  are 

W 
content  to  write  —  for  mass  without  giving  the  unit  a  definite 

«7 

name.  This  is  equivalent  to  saying  that  the  mass  of  a  body, 
expressed  in  sluggs,  is  equal  to  its  weight,  at  London,  expressed  in 
pounds,  divided  by  32.191. 

After  careful  consideration  of  the  foregoing  discussion,  it  will 
be  evident  that  the  engineer's  length-force-time  system  is  exactly 
equivalent  to  a  length-mass-time  system  in  which  the  unit  of 
length  is  the  foot,  the  unit  of  mass  is  the  slugg  or  32.191  pounds' 
mass,  and  the  unit  of  time  is  the  mean  solar  second.  In  the  latter 
system  the  fundamental  units  are  of  the  same  kind  as  those  of 
the  c.g.s.  system.  Hence,  if  the  conversion  factor  for  the  unit 
of  mass  is  taken  as  the  ratio  of  the  magnitude  of  the  slugg  to  that 
of  the  gram,  quantities  expressed  in  the  units  of  the  engineer's 
system  may  be  reduced  to  the  equivalent  values  in  the  c.g.s. 
system  by  the  method  described  at  the  beginning  of  this  article. 


10  THE  THEORY  OF  MEASUREMENTS        [ART.  6 

When,  as  is  frequently  the  case,  the  engineer's  results  are  expressed 
in  terms  of  the  local  weight  of  a  pound  as  a  unit  of  force  in  place 
of  the  pound's  weight  at  London,  the  result  of  a  transformation 
of  units,  carried  out  as  above,  will  be  in  error  by  a  factor  equal  to 
the  ratio  of  the  acceleration  due  to  gravity  at  London  and  at  the 
location  of  the  measurements.  Unless  the  local  gravitational 
acceleration  is  definitely  stated  by  the  observer  and  unless  he 
has  used  his  length-force-time  units  in  a  consistent  manner,  it  is 
impossible  to  derive  the  exact  equivalent  of  his  results  on  the 
c.g.s.  system. 


CHAPTER  II. 
MEASUREMENTS. 

IN  article  two  of  the  last  chapter  we  defined  the  term  "  measure- 
ment "  and  showed  that  any  magnitude  may  be  represented  by 
the  product  of  two  factors,  the  numeric  and  the  unit.  The  object 
of  all  measurements  is  the  determination  of  the  numeric  that  ex- 
presses the  magnitude  of  the  observed  quantity  in  terms  of  the 
chosen  unit.  For  convenience  of  treatment,  they  may  be  classified 
according  to  the  nature  of  the  measured  quantity  and  the  methods 
of  observation  and  reduction. 

7.  Direct  Measurements.  —  The  determination  of  a  desired 
numeric  by  direct  observation  of  the  measured  quantity,  with  the 
aid  of  a  divided  scale  or  other  indicating  device  graduated  in 
terms  of  the  chosen  unit,  is  called  a  direct  measurement. 

Such  measurements  are  possible  when  the  chosen  unit,  together 
with  its  multiples  and  submultiples,  can  be  represented  by  a 
material  standard,  so  constructed  that  it  can  be  directly  applied 
to  the  measured  quantity  for  the  purpose  of  comparison,  or  when 
the  unit  and  the  measured  magnitudes  produce  proportional 
effects  on  a  suitable  indicating  device. 

Lengths  may  be  directly  measured  with  a  graduated  scale, 
masses  by  comparison  with  a  set  of  standard  masses  on  an  equal 
arm  balance,  time  intervals  by  the  use  of  a  clock  regulated  to 
give  mean  solar  time,  and  forces  with  the  aid  of  a  spring  balance. 
Hence  magnitudes  expressible  in  terms  of  the  fundamental  units 
of  either  the  c.g.s.  or  the  engineer's  system  may  be  directly 
measured. 

Many  quantities  expressible  in  terms  of  derived  units,  that  can 
be  represented  by  material  standards,  are  commonly  determined 
by  direct  measurement.  As  illustrations,  we  may  cite  the  deter- 
mination of  the  volume  of  a  liquid  with  a  graduated  flask  and  the 
measurement  of  the  electrical  resistance  of  a  wire  by  comparison 
with  a  set  of  standard  resistances. 

8.  Indirect  Measurements.  —  The  determination  of  a  desired 
numeric    by  computation   from  the  numerics  of   one   or  more 

11 


12  THE  THEORY  OF  MEASUREMENTS         [ART.  9 

directly  measured  magnitudes,  that  bear  a  known  relation  to  the 
desired  quantity,  is  called  an  indirect  measurement. 

The  relation  between  the  observed  and  computed  magnitudes 
may  be  expressed  in  the  general  form 

y  =  Ffa,  Xz,  x3,  .  .  .  a,  b,  c  .  .  .  ), 


where  y,  xt,  x2,  etc.,  represent  measured  or  computed  magnitudes, 
or  the  numerics  corresponding  to  them,  a,  b,  c,  etc.,  represent 
constants,  and  F  indicates  that  there  is  a  functional  relation 
between  the  other  quantities.  This  expression  is  read,  y  equals 
some  function  of  xi,  x*,  etc.,  and  a,  b,  c,  etc.  In  any  particular 
case,  the  form  of  the  function  F  and  the  number  and  nature  of  the 
related  quantities  must  be  known  before  the  computation  of  the 
unknown  quantities  is  undertaken. 

Most  of  the  indirect  measurements  made  by  physicists  and 
engineers  fall  into  one  or  another  of  three  general  classes,  char- 
acterized by  the  nature  of  the  unknown  and  measured  magnitudes 
and  the  form  of  the  function  F. 

9.  Classification  of  Indirect  Measurements. 
I. 

In  the  first  class,  y  represents  the  desired  numeric  of  a  magni- 
tude that  is  not  directly  measured,  either  because  it  is  impossible 
or  inconvenient  to  do  so,  or  because  greater  precision  can  be  at- 
tained by  indirect  methods.  The  form  of  the  function  F  and  the 
numerical  values  of  all  of  the  constants  a,  6,  c,  etc.,  appearing  in 
it,  are  given  by  theory.  The  quantities  xi,  Xz,  etc.,  represent 
the  numerics  of  directly  measured  magnitudes.  In  the  following 
pages  indirect  measurements  belonging  to  this  class  will  sometimes 
be  referred  to  as  derived  measurements. 

As  an  illustration  we  may  cite  the  determination  of  the  density 
s  of  a  solid  sphere  from  direct  measurements  of  its  mass  M  and 
its  diameter  D  with  the  aid  of  the  relation 

M 

=  F^' 

Comparing  this  expression  with  the  general  formula  given  above, 
we  note  that  s  corresponds  to  y,  M  to  xi,  D  to  xa,  J  to  a,  TT  to  6, 

and  that  F  represents  the  function  y^^.    The  form  of  the  func- 


ART.  9]  MEASUREMENTS  13 

tion  is  given  by  the  definition  of  density  as  the  ratio  of  the  mass 
to  the  volume  of  a  body  and  the  numerical  constants  £  and  w  are 
given  by  the  known  relation  between  the  volume  and  diameter  of 
a  sphere. 

II. 

In  the  second  class  of  indirect  measurements,  the  numerical 
constants  a,  b,  c,  etc.,  are  the  unknown  quantities  to  be  computed, 
the  form  of  the  function  F  is  known,  and  all  of  the  quantities  y, 
Xi,  xz,  etc.,  are  obtained  by  direct  measurements  or  given  by 
theory.  The  functions  met  with  in  this  class  of  measurements 
usually  represent  a  continuous  variation  of  the  quantity  y  with 
respect  to  the  quantities  x\,  x2,  etc.,  as  independent  variables. 
Hence  the  result  of  a  direct  measurement  of  y  will  depend  on  the 
particular  values  of  Xi,  x2,  etc.,  that  obtain  at  the  time  of  the 
measurement.  Consequently,  in  computing  the  constants  a,  b,  c, 
etc.,  we  must  be  careful  to  use  only  corresponding  values  of  the 
measured  quantities,  i.e.,  values  that  are,  or  would  be,  obtained 
by  coincident  observations  on  the  several  magnitudes. 

Every  set  of  corresponding  values  of  the  variables  y,  Xi,  x2,  etc., 
when  used  in  connection  with  the  given  function,  gives  an  algebraic 
relation  between  the  unknown  quantities  a,  b,  c,  etc.,  involving 
only  numerical  coefficients  and  absolute  terms.  When  we  have 
obtained  as  many  independent  equations  as  there  are  unknown 
quantities,  the  latter  may  be  determined  by  the  usual  algebraic 
methods.  We  shall  see,  however,  that  more  precise  results  can 
be  obtained  when  the  number  of  independent  measurements  far 
exceeds  the  minimum  limit  thus  set  and  the  computation  is  made 
by  special  methods  to  be  described  hereafter. 

The  determination  of  the  initial  length  L0  and  the  coefficient  of 
linear  expansion  a  of  a  metallic  bar  from  a  series  of  measurements 
of  the  lengths  Lt  corresponding  to  different  temperatures  t  with  the 
aid  of  the  functional  relation 

Lt  =  Lo  (1  +  at) 

is  an  example  of  the  class  of  measurements  here  considered.  Such 
measurements  are  sometimes  called  determinations  of  empirical 
constants. 


14 


THE   THEORY  OF  MEASUREMENTS        [ART.  9 


III. 

The  third  class  of  indirect  measurements  includes  all  cases  in 
which  each  of  a  number  of  directly  measured  quantities  yi,  y*,  ys, 
etc.,  is  a  given  function  of  the  unknown  quantities  Xi,  x2,  XB,  etc., 
and  certain  known  numerical  constants  a,  6,  c,  etc.  In  such  cases 
we  have  as  many  equations  of  the  form 

y1  =  FI  (xi,  x2,  £3,  .  .  .  a,  6,  c,  .  .  .  ), 
2/2  =  F2  (xi,  z2,  $t,  .  .  .  a,  M,  .  .  .  )> 


as  there  are  measured  quantities  yi,  y2,  etc.     This  number  must 
be  at  least  as  great  as  the  number  of  unknowns  Xi,  x2,  etc.,  and 

may  be  much  greater. 
The  functions  Flt  F2, 
etc.,  are  frequently  dif- 
ferent in  form  and  some 
of  them  may  not  con- 
tain all  of  the  un- 
knowns. The  numeri- 
cal constants,  appearing 
in  different  functions, 
are  generally  different. 
But  the  form  of  each 
of  the  functions  and 
the  values  of  all  of  the 
constants  must  be 
known  before  a  solu- 
tion of  the  problem  is 
possible. 

Problems  of  this  type 
are  frequently  met  with 
in  astronomy  and  geod- 
esy. One  of  the  simplest  is  known  as  the  adjustment  of  the 
angles  about  a  point.  Thus,  let  it  be  required  to  find  the  most 
probable  values  of  the  angles  Xi,  x2,  and  x3,  Fig.  1,  from  direct 
measurements  of  yi,  y2,  y3)  .  .  .  y&.  In  this  case  the  general 
equations  take  the  form 


FIG. 


ART.  11]  MEASUREMENTS  15 

2/i  =  xi, 

2/2  =  xi  +  x2, 


2/4  =  X2, 
2/5  =  £2 
2/6  =  »«, 

and  all  of  the  numerical  constants  are  either  unity  or  zero.  The 
solution  of  such  problems  will  be  discussed  in  the  chapter  on  the 
method  of  least  squares. 

10.  Determination  of  Functional  Relations.  —  When  the  form 
of  the  functional  relation  between  the  observed  and  unknown 
magnitudes  is  not  known,  the  solution  of  the  problem  requires 
something  more  than  measurement  and  computation.     In  some 
cases  a  study  of  the  theory  of  the  observed  phenomena,  in  con- 
nection with  that  of  allied  phenomena,  will  suggest  the  form  of  the 
required  function.     Otherwise,  a  tentative  form  must  be  assumed 
after  a  careful  study  of  the  observations  themselves,  generally  by 
graphical  methods.     In  either  case  the  constants  of  the  assumed 
function  must  be  determined  by  indirect  measurements  and  the 
results  tested  by  a  comparison  of  the  observed  and  the  computed 
values  of  the  related  quantities.     If  these  values  agree  within  the 
accidental  errors  of  observation,  the  assumed  function  may  be 
adopted  as  an  empirical  representation  of  the  phenomena.     If 
the  agreement  is  not  sufficiently  close,  the  form  of  the  function 
is  modified,  in  a  manner  suggested  by  the  observations,  and  the 
process  of  computation  and  comparison  is  repeated  until  a  satis- 
factory agreement  is  obtained.     A  more  detailed  treatment  of 
such  processes  will  be  found  in  Chapter  XIII. 

11.  Adjustment,  Setting,  and  Observation  of  Instruments.  — 
Most  of  the  magnitudes  dealt  with  in  physics  and  engineering 
are  determined  by  indirect  measurements.     But  we  have  seen 
that  all  such  quantities  are  dependent  upon  and  computed  from 
directly  measured   quantities.      Consequently,   a   study  of  the 
methods  and  precision  of  direct  measurement  is  of  fundamental 
importance. 

In  general,  every  direct  measurement  involves  three  distinct 
operations.     First:   the  instrument  adopted  is  so  placed  that  its 


16  THE   THEORY  OF  MEASUREMENTS       [ART.  12 

scale  is  in  the  proper  position  relative  to  the  magnitude  to  be 
measured  and  all  of  its  parts  operate  smoothly  in  the  manner  and 
direction  prescribed  by  theory.  Operations  of  this  nature  are 
called  adjustments.  Second:  the  reference  line  of  the  instru- 
ment is  moved,  or  allowed  to  move,  in  the  manner  demanded  by 
theory,  until  it  coincides  with  a  mark  chosen  as  a  point  of  reference 
on  the  measured  magnitude.  We  shall  refer  to  this  operation  as  a 
setting  of  the  instrument.  Third:  the  position  of  the  index  of 
the  instrument,  with  respect  to  its  graduated  scale,  is  read.  This 
is  an  observation. 

As  an  illustration,  consider  the  measurement  of  the  normal 
distance  between  two  parallel  lines  with  a  micrometer  microscope. 
The  instrument  must  be  so  mounted  that  it  can  be  rigidly  clamped 
in  any  desired  position  or  moved  freely  in  the  direction  of  its 
optical  axis  without  disturbing  the  direction  of  the  micrometer 
screw.  The  following  adjustments  are  necessary:  the  axis  of  the 
micrometer  screw  must  be  made  parallel  to  the  plane  of  the  two 
lines  and  perpendicular  to  a  normal  plane  through  one  of  them; 
the  eyepiece  must  be  so  placed  that  the  cross-hairs  are  sharply 
defined;  the  microscope  must  be  moved,  in  the  direction  of  its 
optical  axis,  until  the  image  of  the  two  lines,  or  one  of  them  if  the 
normal  distance  between  them  is  greater  than  the  field  of  view 
of  the  microscope,  is  in  the  same  plane  with  the  cross-hairs.  The 
latter  adjustment  is  correct  when  there  is  no  parallax  between  the 
image  of  the  lines  and  the  cross-hairs.  The  setting  is  made  by 
turning  the  micrometer  head  until  the  intersection  of  the  cross- 
hairs bisects  the  image  of  one  of  the  lines.  Finally  the  reading 
of  the  micrometer  scale  is  observed.  A  similar  setting  and  ob- 
servation are  made  on  the  other  line  and  the  difference  between 
the  two  observations  gives  the  normal  distance  between  the  two 
lines  in  terms  of  the  scale  of  the  micrometer. 

12.  Record  of  Observations.  —  In  the  preceding  article,  the 
word  "observation"  is  used  in  a  very  much  restricted  sense  to 
indicate  merely  the  scale  reading  of  a  measuring  instrument. 
This  restriction  is  convenient  in  dealing  with  the  technique  of 
measurement,  but  many  other  circumstances,  affecting  the  accu- 
racy of  the  result,  must  be  observed  and  taken  into  account  in  a 
complete  study  of  the  phenomena  considered.  There  is,  however^ 
little  danger  of  confusion  in  using  the  word  in  the  two  different 
senses  since  the  more  restricted  meaning  is  in  reality  only  a 


ART.  13]  MEASUREMENTS  17 

special  case  of  the  general.  The  particular  significance  intended 
in  any  special  case  is  generally  clear  from  the  context. 

The  first  essential  for  accurate  measurements  is  a  clear  and 
orderly  record  of  all  of  the  observations.  The  record  should  begin 
with  a  concise  description  of  the  magnitude  to  be  measured,  and 
the  instruments  and  methods  adopted  for  the  purpose.  Instru- 
ments may  frequently  be  described,  with  sufficient  precision,  by 
stating  their  name  and  number  or  other  distinguishing  mark. 
Methods  are  generally  specified  by  reference  to  theoretical  treatises 
or  notes.  The  adjustment  and  graduation  of  the  instruments 
should  be  clearly  stated.  The  date  on  which  the  work  is  carried 
out  and  the  location  of  the  apparatus  should  be  noted. 

Observations,  in  the  restricted  sense,  should  be  neatly  arranged 
in  tabular  form.  The  columns  of  the  table  should  be  so  headed, 
and  referred  to  by  subsidiary  notes,  that  the  exact  significance  of 
all  of  the  recorded  figures  will  be  clearly  understood  at  any  future 
time.  All  circumstances  likely  to  affect  the  accuracy  of  the 
measurements  should  be  carefully  observed  and  recorded  in  the 
table  or  in  suitably  placed  explanatory  notes. 

Observations  should  be  recorded  exactly  as  taken  from  the 
instruments  with  which  they  are  made,  without  mental  computa- 
tion or  reduction  of  any  kind  even  the  simplest.  For  example: 
when  a  micrometer  head  is  divided  into  any  number  of  parts 
other  than  ten  or  one  hundred,  it  is  better  to  use  two  columns  in 
the  table  and  record  the  reading  of  the  main  scale  in  one  and 
that  of  the  micrometer  head  in  the  other  than  to  reduce  the  head 
reading  to  a  decimal  mentally  and  enter  it  in  the  same  column 
with  the  main  scale  reading.  This  is  because  mistakes  are  likely 
to  be  made  in  such  mental  calculations,  even  by  the  most  expe- 
rienced observers,  and,  when  the  final  reduction  of  the  observations 
is  undertaken  at  a  future  time,  it  is  frequently  difficult  or  impos- 
sible to  decide  whether  a  large  deviation  of  a  single  observation 
from  the  mean  of  the  others  is  due  to  an  accidental  error  of  obser- 
vation or  to  a  mistake  in  such  a  mental  calculation. 

13.  Independent,  Dependent,  and  Conditioned  Measure- 
ments. —  Measurements  on  the  same  or  different  magnitudes  are 
said  to  be  independent  when  both  of  the  following  specifications 
are  fulfilled:  first,  the  measured  magnitudes  are  not  required  to 
satisfy  a  rigorous  mathematical  relation  among  themselves; 
second,  the  same  observation  is  not  used  in  the  computation  of 


18  THE  THEORY  OF  MEASUREMENTS       [ART.  14 

any  two  of  the  measurements  and  the  different  observations  are 
entirely  unbiased  by  one  another. 

When  the  first  of  these  specifications  is  fulfilled  and  the  second 
is  not,  the  measurements  are  said  to  be  dependent.  Thus,  when 
several  measurements  of  the  length  of  a  line  are  all  computed 
from  the  same  zero  reading  of  the  scale  used,  they  are  all  dependent 
on  that  observation  and  any  error  in  the  position  of  the  zero  mark 
affects  all  of  them  by  exactly  the  same  amount.  When  the  position 
of  the  index  relative  to  the  scale  of  the  measuring  instrument  is 
visible  while  the  settings  are  being  made,  there  is  a  marked  tendency 
to  set  the  instrument  so  that  successive  observations  will  be  exactly 
alike  rather  than  to  make  an  independent  judgment  of  the  bisection 
of  the  chosen  mark  in  each  case.  The  observations,  corresponding 
to  settings  made  in  this  manner,  are  biased  by  a  preconceived 
notion  regarding  the  correct  position  of  the  index  and  the  measure- 
ments computed  from  them  are  not  independent.  The  impor- 
tance of  avoiding  faulty  observations  of  this  type  cannot  be  too 
strongly  emphasized.  They  not  only  vitiate  the  results  of  our 
measurements,  but  also  render  a  determination  of  their  precision 
impossible. 

Measurements  that  do  not  satisfy  the  first  of  the  above  speci- 
fications are  called  conditioned  measurements.  The  different 
determinations  of  each  of  the  related  quantities  may  or  may  not 
be  independent,  according  as  they  do  or  do  not  satisfy  the  second 
specification,  but  the  adjusted  results  of  all  of  the  measurements 
must  satisfy  the  given  mathematical  relation.  Thus,  we  may 
make  a  number  of  independent  measurements  of  each  of  the 
angles  of  a  plane  triangle,  but  the  mean  results  must  be  so  adjusted 
that  the  sum  of  the  accepted  values  is  equal  to  one  hundred  and 
eighty  degrees. 

14.  Errors  and  the  Precision  of  Measurements.  —  Owing  to 
unavoidable  imperfections  and  lack  of  constant  sensitiveness  in 
our  instruments,  and  to  the  natural  limit  to  the  keenness  of  our 
senses,  the  results  of  our  observations  and  measurements  differ 
somewhat  from  the  true  numeric  of  the  observed  magnitude. 
Such  differences  are  called  errors  of  observation  or  measurement. 
Some  of  them  are  due  to  known  causes  and  can  be  eliminated, 
with  sufficient  accuracy,  by  suitable  computations.  Others  are 
apparently  accidental  in  nature  and  arbitrary  in  magnitude. 
Their  probable  distribution,  in  regard  to  magnitude  and  frequency 


ART.  15]  MEASUREMENTS  19 

of  occurrence,  can  be  determined  by  statistical  methods  when  a 
sufficient  number  of  independent  measurements  is  available. 

The  precision  of  a  measurement  is  the  degree  of  approximation 
with  which  it  represents  the  true  numeric  of  the  observed  magni- 
tude. Usually  our  measurements  serve  only  to  determine  the 
probable  limits  within  which  the  desired  numeric  lies.  Looked 
at  from  this  point  of  view,  the  precision  of  a  measurement  may  be 
considered  to  be  inversely  proportional  to  the  difference  between 
the  limits  thus  determined.  It  increases  with  the  accuracy, 
adaptability,  and  sensitiveness  of  the  instruments  used,  and  with 
the  skill  and  care  of  the  observer.  But,  after  a  very  moderate 
precision  has  been  attained,  the  labor  and  expense  necessary  for 
further  increase  is  very  great  in  proportion  to  the  result  obtained. 

A  measurement  is  of  little  practical  value  unless  we  know  the 
precision  with  which  it  represents  the  observed  magnitude. 
Hence  the  importance  of  a  thorough  study  of  the  nature  and  dis- 
tribution of  errors  in  general  and  of  the  particular  errors  that 
characterize  an  adopted  method  of  measurement.  At  first  sight 
it  might  seem  incredible  that  such  errors  should  follow  a  definite 
mathematical  law.  But,  when  the  number  of  observations  is 
sufficiently  great,  we  shall  see  that  the  theory  of  probability  leads 
to  a  definite  and  easily  calculated  measure  of  the  precision  of  a 
single  observation  and  of  the  result  computed  from  a  number 
of  observations. 

15.  Use  of  Significant  Figures.  —  When  recording  the  nu- 
merical results  of  observations  or  measurements,  and  during  all 
of  the  necessary  computations,  the  number  of  significant  figures 
employed  should  be  sufficient  to  express  the  attained  precision 
and  no  more.  By  significant  figures  we  mean  the  nine  digits  and 
zeros  when  not  used  merely  to  locate  the  decimal  point. 

In  the  case  of  the  direct  observation  of  the  indications  of  instru- 
ments, the  above  specification  is  usually  sufficiently  fulfilled  by 
allowing  the  last  recorded  significant  figure  to  represent  the 
estimated  tenth  of  the  smallest  division  of  the  graduated  scale. 
For  example:  in  measuring  the  length  of  a  line,  with  a  scale 
divided  in  millimeters,  the  position  of  the  ends  of  the  line  would 
be  recorded  to  the  nearest  estimated  tenth  of  a  millimeter. 

Generally,  computed  results  should  be  so  recorded  that  the 
limiting  values,  used  to  express  the  attained  precision,  differ  by 
only  a  few  units  in  the  last  one  or  two  significant  figures.  Thus: 


20  THE  THEORY  OF  MEASUREMENTS       [ART.  15 

if  the  length  of  a  line  is  found  to  lie  between  15.65  millimeters  and 
15.72  millimeters,  we  should  write  15.68  millimeters  as  the  result 
of  our  measurement.  The  use  of  a  larger  number  of  significant 
figures  would  be  not  only  a  waste  of  space  and  labor,  but  also  a 
false  representation  of  the  precision  of  the  result.  Most  of  the 
magnitudes  we  are  called  upon  to  measure  are  incommensurable 
with  the  chosen  unit,  and  hence  there  is  no  limit  to  the  number 
of  significant  figures  that  might  be  used  if  we  chose  to  do  so;  but 
experienced  observers  are  always  careful  to  express  all  observa- 
tions and  results  and  carry  out  all  computations  with  a  number 
just  sufficient  to  represent  the  attained  precision.  The  use  of 
too  many  or  too  few  significant  figures  is  strong  evidence  of  inex- 
perience or  carelessness  in  making  observations  and  computations. 
More  specific  rules  for  determining  the  number  of  significant 
figures  to  be  used  in  special  cases  will  be  developed  in  connection 
with  the  methods  for  determining  the  precision  of  measurements. 

The  number  of  significant  figure^  in  any  numerical  expression 
is  entirely  independent  of  the  position  of  the  decimal  point. 
Thus:  each  of  the  numbers  5,769,600,  5769,  57.69,  and  0.0005769 
is  expressed  by  four  significant  figures  and  represents  the  corre- 
sponding magnitude  within  one-tenth  of  one  per  cent,  notwith- 
standing the  fact  that  the  different  numbers  correspond  to  differ- 
ent magnitudes.  In  general,  the  location  of  the  decimal  point 
shows  the  order  of  magnitude  of  the  quantity  represented  and 
the  number  of  significant  figures  indicates  the  precision  with  which 
the  actual  numeric  of  the  quantity  is  known. 

In  writing  very  large  or  very  small  numbers,  it  is  convenient 
to  indicate  the  position  of  the  decimal  point  by  means  of  a  positive 
or  negative  power  of  ten.  Thus:  the  number  56,400,000  may 
be  written  564  X  105  or,  better,  5.64  X  107,  and  0.000075  may 
be  written  75  X  W~«  or  7.5  X  10~5.  When  a  large  number  of 
numerical  observations  or  results  are  to  be  tabulated  or  used  in 
computation,  a  considerable  amount  of  time  and  space  is  saved 
by  adopting  this  method  of  representation.  The  second  of  the 
two  forms,  illustrated  above,  is  very  convenient  in  making  com- 
putations by  means  of  logarithms,  as  in  this  case  the  power  of 
ten  always  represents  the  characteristic  of  the  logarithm  of  the 
corresponding  number. 

In  rounding  numbers  to  the  required  number  of  significant 
figures,  the  digit  in  the  last  place  held  should  be  increased  by  one 


ART.  17]  MEASUREMENTS  21 

unit  when  the  digit  in  the  next  lower  place  is  greater  than  five, 
and  left  unchanged  when  the  neglected  part  is  less  than  five- 
tenths  of  a  unit.  When  the  neglected  part  is  exactly  five-tenths 
of  a  unit  the  last  digit  held  is  increased  by  one  if  odd,  and  left 
unchanged  if  even.  Thus:  5687.5  would  be  rounded  to  5688  and 
5686.5  to  5686. 

1 6.  Adjustment    of   Measurements.  —  The  results  of   inde- 
pendent measurements  of  the  same  magnitude  by  the  same  or 
different  methods  seldom  agree  with  one  another.     This  is  due  to 
the  fact  that  the  probability  for  the  occurrence  of  errors  of  exactly 
the  same  character  and  magnitude  in  the  different  cases  is  very 
small  indeed.     Hence  we  are  led  to  the  problem  of  determining 
the  best  or  most  probable  value  of  the  numeric  of  the  observed 
magnitude  from  a  series  of  discordant  measurements.     The  given 
data  may  be  all  of  the  same  precision  or  it  may  be  necessary  to 
assign  a  different  degree  of  accuracy  to  the  different  measure- 
ments.    In  either  case  the  solution  of  the  problem  is  called  the 
adjustment  of  the  measurements. 

The  principle  of  least  squares,  developed  in  the  theory  of  errors 
that  leads  to  the  measure  of  precision  cited  above,  is  the  basis 
of  all  such  adjustments.  But  the  particular  method  of  solution 
adopted  in  any  given  case  depends  on  the  nature  of  the  measure- 
ments considered.  In  the  case  of  a  series  of  direct,  equally  pre- 
cise, measurements  of  a  single  quantity,  the  principle  of  least 
squares  leads  to  the  arithmetical  mean  as  the  most  probable,  and 
therefore  the  best,  value  to  assign  to  the  measured  quantity. 
This  is  also  the  value  that  has  been  universally  adopted  on  a  priori 
grounds.  In  fact  many  authors  assume  the  maximum  probability 
of  the  arithmetical  mean  as  the  axiomatic  basis  for  the  develop- 
ment of  the  law  of  errors. 

The  determination  of  empirical  relations  between  measured 
quantities  and  the  constants  that  enter  into  them  is  also  based 
on  the  principle  of  least  squares.  For  this  reason,  such  deter- 
minations are  treated  in  connection  with  the  discussion  of  the 
methods  for  the  adjustment  of  measurements. 

17.  Discussion  of  Instruments  and  Methods.  —  The  theory 
of  errors  finds  another  very  important  application  in  the  discussion 
of  the  relative  availableness  and  accuracy  of  different  instruments 
and  methods  of  measurement.     Used  in  connection  with  a  few 
preliminary   measurements   and   a   thorough   knowledge   of   the 


22  THE   THEORY  OF  MEASUREMENTS       [ART.  17 

theory  of  the  proposed  instruments  and  methods,  it  is  sufficient 
for  the  determination  of  the  probable  precision  of  an  extended 
series  of  careful  observations.  By  such  means  we  are  able  to 
select  the  instruments  and  methods  best  adapted  to  the  particular 
purpose  in  view.  We  also  become  acquainted  with  the  parts  of 
the  investigation  that  require  the  greatest  skill  and  care  in  order 
to  give  a  result  with  the  desired  precision. 

The  cost  of  instruments  and  the  time  and  skill  required  in 
carrying  out  the  measurements  increase  much  more  rapidly  than 
the  corresponding  precision  of  the  results.  Hence  these  factors 
must  be  taken  into  account  in  determining  the  availableness  of  a 
proposed  method.  It  is  by  no  means  always  necessary  to  strive 
for  the  greatest  attainable  precision.  In  fact,  it  would  be  a 
waste  of  time  and  money  to  carry  out  a  given  measurement  with 
greater  precision  than  is  required  for  the  use  to  which  it  is  to  be 
put.  For  many  practical  purposes,  a  result  correct  within  one- 
tenth  of  one  per  cent,  or  even  one  per  cent,  is  amply  sufficient. 
In  such  cases  it  is  essential  to  adopt  apparatus  and  methods  that 
will  give  results  definitely  within  these  limits  without  incurring 
the  greater  cost  and  labor  necessary  for  more  precise  deter- 
minations. 


CHAPTER   III. 
CLASSIFICATION   OF  ERRORS. 

ALL  measurements,  of  whatever  nature,  are  subject  to  three 
distinct  classes  of  errors,  namely,  constant  errors,  mistakes,  and 
accidental  errors. 

18.  Constant  Errors.  —  Errors  that  can  be  determined  in 
sign  and  magnitude  by  computations  based  on  a  theoretical 
consideration  of  the  method  of  measurement  used  or  on  a  pre- 
liminary study  and  calibration  of  the  instruments  adopted  are 
called  constant  errors.  They  are  sometimes  due  to  inadequacy  of 
an  adopted  method  of  measurement,  but  more  frequently  to 
inaccurate  graduation  and  imperfect  adjustment  of  instruments. 

As  a  simple  illustration,  consider  the  measurement  of  the 
length  of  a  straight  line  with  a  graduated  scale.  If  the  scale  is 
not  held  exactly  parallel  to  the  line,  the  result  will  be  too  great 
or  too  small  according  as  the  line  of  sight  in  reading  the  scale  is 
normal  to  the  line  or  to  the  scale.  The  magnitude  of  the  error 
thus  introduced  depends  on  the  angle  between  the  line  and  the 
scale  and  can  be  exactly  computed  when  we  know  this  angle  and 
the  circumstances  of  the  observations.  If  the  scale  is  not  straight, 
if  its  divisions  are  irregular,  or  if  they  are  not  of  standard  length, 
the  result  of  the  measurement  will  be  in  error  by  an  amount 
depending  on  the  magnitude  and  distribution  of  these  inaccuracies 
of  construction.  The  sign  and  magnitude  of  such  errors  can 
genero1ly  be  determined  by  a  careful  study  and  calibration  of  the 
scai 

If  M  represents  the  actual  numeric  of  the  measured  magnitude, 
MQ  the  observed  numeric,  and  Ci,  C2,  C3,  etc.,  the  constant  errors 
inherent  in  the  method  of  measurement  and  the  instruments  used, 

M  =  Mo  +  Ci  +  C2  +  C3  +  •  •  -  .  (1) 

The  necessary  number  of  correction  terms  Ci,  G'2,  Cz,  etc.,  is 
determined  by  a  careful  study  of  the  theory  and  practical  appli- 
cation of  the  apparatus  and  method  used  in  finding  MQ.  The 
magnitude  and  sign  of  each  term  are  determined  by  subsidiary 

23 


24  THE   THEORY  OF  MEASUREMENTS       [ART.  18 

measurements  or  calculated,  on  theoretical  grounds,  from  known 
data.  Thus,  in  the  above  illustration,  suppose  that  the  scale  is 
straight  and  uniformly  graduated,  that  each  of  its  divisions  is 
1.01  times  as  long  as  the  unit  in  which  it  is  supposed  to  be  gradu- 
ated, and  that  the  line  of  the  graduations  makes  an  angle  a  with 
the  line  to  be  measured.  Under  these  conditions,  the  number  of 
correction  terms  reduces  to  two:  the  first,  Ci,  due  to  the  false 
length  of  the  scale  divisions,  and  the  second,  C2,  due  to  the  lack 
of  parallelism  between  the  scale  and  the  line. 

Since  the  actual  length  of  each  division  is  1.01,  the  .length  of 
Mo  divisions,  i.e.,  the  length  that  would  have  been  observed  on 
an  accurate  scale,  is 

Ml  =  Mo  X  1.01  =  Mo  +  0.01  Mo  =  Mo  +  Ci, 
...     Ci  =  +  0.01  Mo. 

If  the  line  of  sight  is  normal  to  the  line  in  making  the  observa- 
tions, the  length  M2  that  would  have  been  obtained  if  the  scale 
had  been  properly  placed  is 

M2  =  MO  cos  a  =  MO  +  Czj 
/.     C2  =-M0(l-cosa)=-2M0sin2^ 
and  (1)  takes  the  form 

M=  Mo  +  0.01  Mo  -  2M0sin2|> 

=  M0(l+0.01-2sin2^Y 

The  precision  with  which  it  is  necessary  to  determine  the  cor- 
rection terms  Ci,  C2,  etc.,  and  frequently  the  number  of  these 
terms  that  should  be  employed  depends  on  the  precision  with 
which  the  observed  numeric  M0  is  determined.  If  M0  is  measured 
within  one-tenth  of  one  per  cent  of  its  magnitude,  the  several 
correction  terms  should  be  determined  within  one  one-hundredth 
of  one  per  cent  of  M0,  in  order  that  the  neglected  part  of  the  sum 
of  the  corrections  may  be  less  than  one-tenth  of  one  per  cent  of 
M0.  If  any  correction  term  is  found  to  be  less  than  the. above 
limit,  it  may  be  neglected  entirely  since  it  is  obviously  useless 
to  apply  a  correction  that  is  less  than  one-tenth  of  the  uncer- 
tainty of  M0. 

In  our  illustration,  suppose  that  the  precision  is  such  that  we 
are  sure  that  M0  is  less  than  1.57  millimeters  and  greater  than 


ART.  19]  CLASSIFICATION   OF   ERRORS  25 

1.55  millimeters,  but  is  not  sufficient  to  give  the  fourth  significant 
figure  within  several  units.  Obviously,  it  would  be  useless  to 
determine  Ci  and  C%  closer  than  0.001  millimeter,  and  if  the  mag- 
nitude of  either  of  these  quantities  is  less  than  0.001  millimeter 
our  knowledge  of  the  true  value  of  M  is  not  increased  by  making 
the  corresponding  correction.  In  fact,  it  is  usually  impossible 
to  determine  the  C's  with  greater  accuracy  than  the  above  limit, 
since,  as  in  our  illustration,  MQ  is  usually  a  factor  in  the  correction 
terms.  Hence  the  writing  down  of  more  than  the  required  num- 
ber of  significant  figures  is  mere  waste  of  labor. 

When  considering  the  availableness  of  proposed  methods  and 
apparatus,  it  is  important  to  investigate  the  nature  and  magni- 
tude of  the  constant  errors  inherent  in  their  use.  It  sometimes 
happens  that  the  sources  of  such  errors  can  be  sufficiently  elimi- 
nated by  suitable  adjustment  of  the  instruments  or  modification 
of  the  method  of  observation.  When  this  is  not  possible  the 
conditions  should  be  so  chosen  that  the  correction  terms  can  be 
computed  with  the  required  precision.  Even  when  all  possible 
precautions  have  been  taken,  it  very  seldom  happens  that  the 
sum  of  the  constant  errors  reduces  to  zero  or  that  the  magni- 
tude of  the  necessary  corrections  can  be  exactly  determined. 
Moreover,  such  errors  are  never  rigorously  constant,  but  present 
small  fortuitous  variations,  which,  to  some  extent,  are  indistinguish- 
able from  the  accidental  errors  to  be  described  later. 

A  more  detailed  discussion  of  constant  errors  and  the  limits 
within  which  they  should  be  determined  will  be  given  after  we 
have  developed  the  methods  for  estimating  the  precision  of  the 
observed  numeric  M. 

19.  Personal  Errors.  —  When  setting  cross-hairs,  or  any  other 
indicating  device,  to  bisect  a  chosen  mark,  some  observers  will 
invariably  set  too  far  to  one  side  of  the  center,  while  others  will 
as  consistently  set  on  the  other  side.  Again,  in  timing  a  transit, 
some  persons  will  signal  too  soon  and  others  too  late.  With 
experienced  and  careful  observers,  the  errors  introduced  in  this 
manner  are  small  and  nearly  constant  in  magnitude  and  sign, 
but  they  are  seldom  entirely  negligible  when  the  highest  possible 
precision  is  sought. 

Errors  of  this  nature  will  be  called  personal  errors,  since  their 
magnitude  and  sign  depend  on  personal  peculiarities  of  the 
observer.  Their  elimination  may  sometimes  be  effected  by  a 


26  THE  THEORY  OF  MEASUREMENTS        [ART.  20 

careful  study  of  the  nature  of  such  peculiarities  and  the  magnitude 
of  the  effects  produced  by  them  under  the  conditions  imposed 
by  the  particular  problem  considered.  Suitable  methods  for  this 
purpose  are  available  in  connection  with  most  of  the  investiga- 
tions in  which  an  exact  knowledge  of  the  personal  error  is  essential. 
Such  a  study  is  .frequently  referred  to  as  a  determination  of  the 
"Personal  Equation"  of  the  observer. 

20.  Mistakes.  —  Mistakes  are  errors  due  to  reading  the  indi- 
cations of  an  instrument  carelessly  or  to  a  faulty  record  of  the 
observations.     The  most  frequent  of  these   are   the   following : 
the  wrong  integer  is  placed  before  an  accurate  fractional  reading, 
e.g.,  9.68  for  19.68;  the  reading  is  made  in  the  wrong  direction  of 
the  scale,  e.g.,  6.3  for  5.7;  the  significant  figures  of  a  number  are 
transposed,  e.g.,  56  is  written  for  65.     Care  and  strict  attention 
to  the  work  in  hand  are  the  only  safeguards  against  such  mistakes. 

When  a  large  number  of  observations  have  been  systematically 
taken  and  recorded,  it  is  sometimes  possible  to  rectify  an  obvious 
mistake,  but  unless  this  can  be  done  with  certainty  the  offending 
observation  should  be  dropped  from  the  series.  This  statement 
does  not  apply  to  an  observation  showing  a  large  deviation  from 
the  mean  but  only  to  obvious  mistakes. 

21.  Accidental  Errors.  —  When  a  series  of  independent  meas- 
urements of  the  same  magnitude  have  been  made,  by  the  same 
method  and  apparatus  and  with  equal  care,  the  results  generally 
differ  among  themselves  by  several  units  in  the  last  one  or  two 
significant  figures.     If  in  any  case  they  are  found  to  be  identical, 
it  is  probable  that  the  observations  were  not  independent,  the 
instruments  adopted  were  not  sufficiently  sensitive,  the  maximum 
precision  attainable  was   not  utilized,  or  the  observations  were 
carelessly  made.     Exactly  concordant  measurements  are  quite  as 
strong  evidence  of  inaccurate  observation  as  widely  divergent 
ones. 

As  the  accuracy  of  method  and  the  sensitiveness  of  instruments 
is  increased,  the  number  of  concordant  figures  in  the  result  in- 
creases but  differences  always  occur  in  the  last  attainable  figures. 
Since  there  is,  generally,  no  reason  to  suppose  that  any  one  of  the 
measurements  is  more  accurate  than  any  other,  we  are  led  to 
believe  that  they  are  all  affected  by  small  unavoidable  errors. 

After  all  constant  errors  and  mistakes  have  been  corrected,  the  re- 
maining differences  between  the  individual  measurements  and  the  true 


ART.  22]  CLASSIFICATION  OF   ERRORS  27 

numeric  of  the  measured  magnitude  are  called  accidental  errors. 
They  are  due  to  the  combined  action  of  a  large  number  of  inde- 
pendent causes  each  of  which  is  equally  likely  to  produce  a  posi- 
tive or  a  negative  effect.  Probably  most  of  them  have  their 
origin  in  small  fortuitous  variations  in  the  sensitiveness  and 
adjustment  of  our  instruments  and  in  the  keenness  of  our  senses 
of  sight,  hearing,  and  touch.  It  is  also  possible  that  the  correla- 
tion of  our  sense  perceptions  and  the  judgments  that  we  draw 
from  them  are  not  always  rigorously  the  same  under  the  same 
set  of  stimuli. 

Suppose  that  N  measurements  of  the  same  quantity  have  been 
made  by  the  same  method  and  with  equal  care.  Let  ai,  a^,  «3, 
.  .  .  aN  represent  the  several  results  of  the  independent  meas- 
urements, after  all  constant  errors  and  mistakes  have  been  elim- 
inated, and  let  X  represent  the  true  numeric  of  the  measured 
magnitude.  Then  the  accidental  errors  of  the  individual  measure- 
ments are  given  by  the  differences, 

Ai  -  ai  -  X,  A2  =  a2  -  X,  A3  =  a3  -  X}  .  .  .  A^  =  aN-X.  (2) 

The  accidental  errors  AI,  A2,  .  .  .  A#  thus  denned  are  sometimes 
called  the  true  errors  of  the  observations  ai,  a2,  .  .  .  aN. 

22.  Residuals.  —  Since  the  individual  measurements  a\t  a?, 
.  .  .  aN  differ  among  themselves,  and  since  there  is  no  reason  to 
suppose  that  any  one  of  them  is  more  accurate  than  any  other,  it 
is  never  possible  to  determine  the  exact  magnitude  of  the  numeric 
X.  Hence  the  magnitude  of  the  accidental  errors  A  i,  A  2,  .  .  .  A# 
can  never  be  exactly  determined.  But,  if  x  is  the  most  probable 
value  that  we  can  assign  to  the  numeric  X  on  the  basis  of  our 
measurements,  we  can  determine  the  differences 

ri  =  di  —  x,     rz  =  a2  —  x,     .  .  .  rN  =  aN  —  x.  (3) 

These  differences  are  called  the  residuals  of  the  individual  measure- 
ments dij  02,  .  .  .  aN.  They  represent  the  most  probable  values 
that  we  can  assign  to  the  accidental  errors  AI,  A2,  .  .  .  A#  on  the 
basis  of  the  given  measurements. 

It  should  be  continually  borne  in  mind  that  the  residuals  thus 
determined  are  never  identical  with  the  accidental  errors.  How- 
ever precise  our  measurements  may  be,  the  probability  that  x  is 
exactly  equal  to  X  is  always  less  than  unity.  As  the  number 
and  precision  of  measurements  increase,  the  difference  between 


28  THE  THEORY  OF  MEASUREMENTS       [ART.  23 

the  magnitudes  x  and  X  decreases,  and  the  residuals  continually 
approach  the  accidental  errors,  but  exact  equality  is  never  attain- 
able with  a  finite  number  of  observations. 

23.  Principles  of  Probability.  —  The  theory  of  errors  is  an 
application  of  the  principles  of  probability  to  the  discussion  of 
series  of  discordant  measurements  for  the  purpose  of  determining 
the  most  probable  numeric  that  can  be  assigned  to  the  measured 
quantity  and  making  an  estimate  of  the  precision  of  the  result 
thus  obtained.  A  discussion  of  the  fundamental  principles  of 
the  theory  of  probability,  sufficient  for  this  purpose,  is  given  in 
most  textbooks  on  advanced  algebra,  and  the  student  should 
master  them  before  undertaking  the  study  of  the1  theory  of  errors. 

For  the  sake  of  convenience  in  reference,  the  three  most  useful 
propositions  are  stated  below  without  proof. 

PROPOSITION  1.  If  an  event  can  happen  in  n  independent 
ways  and  either  happen  or  fail  in  N  independent  ways,  the  prob- 
ability p  that  it  will  occur  in  a  single  trial  at  random  is  given  by 

the  relation 

n  ,A. 

p  -  r  w 

Also  if  p'  is  the  probability  that  it  will  fail  in  a  single  trial  at 
random, 

p»  =  l_p  =  !_.».  (5) 

PROPOSITION  2.  If  the  probabilities  for  the  separate  occurrence 
of  n  independent  events  are  respectively  pi,  p%,  .  .  .  pn,  the  prob- 
ability PS  that  some  one  of  these  events  will  occur  in  a  single  trial 
at  random  is  given  by  the  relation 

PS  =  Pi  +  Pz  +  Pz  +  '  '  '  +  P^  (6) 

PROPOSITION  3.  If  the  probabilities  for  the  separate  occurrence 
of  n  independent  events  are  respectively  pi,  p2,  .  .  .  pn,  the 
probability  P  that  all  of  the  events  will  occur  at  the  same  time  is 
given  by  the  relation 

P  =  Pi  X  P2  X     •    •    •     X  Pn.  (7) 


CHAPTER  IV. 
THE  LAW  OF  ACCIDENTAL  ERRORS. 

24.  Fundamental  Propositions.  —  The  theory  of  accidental 
errors  is  based  on  the  principle  of  the  arithmetical  mean  and  the 
three  axioms  of  accidental  errors.     When  the  word  "  error  "  is  used 
without  qualification,  in  the  statement  of  these  propositions  and 
in  the  following  pages,  accidental  errors  are  to  be  understood. 

Principle  of  the  Arithmetical  Mean.  —  The  most  probable  value 
that  can  be  assigned  to  the  numeric  of  a  measured  magnitude,  on 
the  basis  of  a  number  of  equally  trustworthy  direct  measurements, 
is  the  arithmetical  mean  of  the  given  'measurements. 

This  proposition  is  self-evident  in  the  case  of  two  independent 
measurements,  made  by  the  same  method  with  equal  care,  since 
one  of  them  is  as  likely  to  be  exact  as  the  other,  and  hence  it  is 
more  probable  that  the  true  numeric  lies  halfway  between  them 
than  in  any  other  location.  Its  extension  to  more  than  two 
measurements  is  the  only  rational  assumption  that  we  can  make 
and  is  sanctioned  by  universal  usage. 

First  Axiom.  —  In  any  large  number  of  measurements,  positive 
and  negative  errors  of  the  same  magnitude  are  equally  likely  to 
occur.  The  number  of  negative  errors  is  equal  to  the  number 
of  positive  errors. 

Second  Axiom.  —  Small  errors  are  much  more  likely  to  occur 
than  large  ones. 

Third  Axiom.  —  All  of  the  errors  of  the  measurements  in  a 
given  series  lie  between  equal  positive  and  negative  limits.  Very 
large  errors  do  not  occur. 

The  foundation  of  these  propositions  is  the  same  as  that  of  the 
axioms  of  geometry.  Namely:  they  are  general  statements  that 
are  admitted  as  self-evident  or  accepted  as  a  basis  of  argument  by 
all  competent  persons.  Their  justification  lies  in  the  fact  that 
the  results  derived  from  them  are  found  to  be  in  agreement  with 
experience. 

25.  Distribution  of  Residuals.  —  It  was  pointed  out  in  article 
twenty-two  that  the  true  accidental  errors,  represented  by  A's, 

29 


30 


THE   THEORY   OF  MEASUREMENTS       [ART.  26 


cannot  be  determined  in  practice,  but  the  residuals,  represented 
by  r's,  can  be  computed  from  the  given  observations  by  equation 
(3).  The  A's  may  be  considered  as  the  limiting  values  toward 
which  the  r's  approach  as  the  number  of  observations  is  indefinitely 
increased.  If  the  residuals  corresponding  to  a  very  large  num- 
ber of  observations  are  arranged  in  groups  according  to  sign  and 
magnitude,  the  groups  containing  very  small  positive  or  negative 
residuals  will  be  found  to  be  the  largest,  and,  in  general,  the  magni- 
tude of  the  groups  will  decrease  nearly  uniformly  as  the  magnitude 
of  the  contained  residuals  increases  either  positively  or  negatively. 
Let  n  represent  the  number  of  residuals  in  any  group,  and  r  their 
common  magnitude,  then  the  distribution  of  the  residuals,  in 
regard  to  sign  and  magnitude,  may  be  represented  graphically 
by  laying  off  ordinates  proportional  to  the  numbers  n  against 


abscissae  proportional  to  the  corresponding  magnitudes  r.  The 
points,  thus  located,  will  be  approximately  uniformly  distributed 
about  a  curve  of  the  general  form  illustrated  in  Fig.  2. 

The  number  of  residuals  in  each  group  will  increase  with  the 
total  number  of  measurements  from  which  the  r's  are  computed. 
Consequently  the  ordinates  of  the  curve  in  Fig.  2  will  depend  on 
the  number  of  observations  considered  as  well  as  on  their  accuracy. 
Hence,  if  we  wish  to  compare  different  series  of  measurements  with 
regard  to  accuracy,  we  must  in  some  way  eliminate  the  effect  of 
differences  in  the  number  of  observations.  Moreover,  we  are  not 
so  much  concerned  with  the  total  number  of  residuals  of  any  given 
magnitude  as  with  the  relative  number  of  residuals  of  different 
magnitudes.  For,  as  we  shall  see,  the  acuracy  of  a  series  of 
observations  depends  on  the  ratio  of  the  number  of  small  errors 
to  the  number  of  large  ones. 

26.  Probability  of  Residuals.  —  Suppose  that  a  very  large 
number  N  of  independent  measurements  have  been  made  and  that 


AKF.27J     THE  LAW  OF  ACCIDENTAL  ERRORS  31 

the  corresponding  residuals  have  been  computed  by  equation  (3). 
By  arranging  the  results  in  groups  according  to  sign  and  magni- 
tude, suppose  we  find  HI  residuals  of  magnitude  n,  n2  of  magni- 
tude r2,  etc.,  and  n\  of  magnitude  —  n,  n/  of  magnitude  —  r2,  etc. 
If  we  choose  one  of  the  measurements  at  random,  the  probability 

that  the  corresponding  residual  is  equal  to  r\  is  -^  ,  since  there 

are  N  residuals  and  n\  of  them  are  equal  to  r\.  In  general,  if  y\,  y2, 
•  •  •  Hi,  2/2',  •  •  •  represent  the  probabilities  for  the  occurrence 
of  residuals  equal  to  n,  r2,  .  .  .  —  n,  —  r2,  .  .  .  respectively, 


When  N  is  increased  by  increasing  the  number  of  measurements, 
each  of  the  n's  is  increased  in  nearly  the  same  ratio  since  the 
residuals  of  the  new  measurements  are  distributed  in  essentially 
the  same  manner  as  the  old  ones,  provided  all  of  the  measure- 
ments considered  are  made  by  the  same  method  and  with  equal 
care.  Consequently,  the  y's  corresponding  to  a  definite  method 
of  observation  are  nearly  independent  of  the  number  of  measure- 
ments. As  N  increases  they  oscillate,  with  continually  decreas- 
ing amplitude,  about  the  limiting  values  that  would  be  obtained 
with  an  infinite  number  of  observations.  Hence  the  form  of  a 
curve,  having  y's  for  ordinates  and  corresponding  r's  for  abscissae, 
depends  on  the  accuracy  of  the  measurements  considered  and  is 
sensibly  independent  of  N,  provided  it  is  a  large  number. 

27.  The  Unit  Error.  —  The  relative  accuracy  of  different 
series  of  measurements  might  be  studied  with  the  aid  of  the  corre- 
sponding y  :  r  curves,  but  since  the  y's  are  abstract  numbers,  and 
the  r's  are  concrete,  being  of  the  same  kind  as  the  measurements, 
it  is  better  to  adopt  a  slightly  different  mode  of  representation. 
For  this  purpose,  each  of  the  r's  is  divided  by  an  arbitrary  con- 
stant k,  of  the  same  kind  as  the  measurements,  and  the  abstract 

numbers  y^>  -^>  etc.,  are  used  as  abscissae  in  place  of  the  r's.     In 

A/       K 

the  following  pages,  k  will  be  called  the  unit  error.  Its  magnitude 
may  be  arbitrarily  chosen  in  particular  cases,  but,  when  not 
definitely  specified  to  the  contrary,  it  will  be  taken  equal  to  the 
least  magnitude  that  can  be  directly  observed  with  the  instru- 
ments and  methods  used  in  making  the  measurements.  To 


32 


THE  THEORY  OF  MEASUREMENTS       I  ART.  28 


illustrate:  suppose  we  are  measuring  a  given  length  with  a  scale 
divided  in  millimeters.  By  estimation,  the  separate  observations 
can  be  made  to  one-tenth  of  a  millimeter.  Hence,  in  this  case 
we  should  take  k  equal  to  one-tenth  of  a  millimeter. 

If  the  residuals  are  arranged  in  the  order  of  increasing  magni- 
tude, it  is  obvious  that  the  successive  differences  TI  —  r0,  r?  —  TI 
etc.,  are  all  equal  to  k.  Hence,  if  the  most  probable  value  of  the 
measured  quantity,  x  in  equation  (3),  is  taken  to  the  same  num- 
ber of  significant  figures  as  the  individual  measurements,  all  of 
the  residuals  are  integral  multiples  of  k  and  we  have 


k 


k 


28.  The  Probability  Curve. —  The  result  of  a  study  of  the 
distribution  of  the  residuals  may  be  arranged  as  illustrated  in  the 
following  table,  where  n  is  the  number  of  residuals  of  magnitude 
r;  y  is  the  probability  that  a  single  residual,  chosen  at  random,  is 
of  magnitude  r;  N  is  the  total  number  of  measurements,  and  k  is 
the  unit  error. 


r 

n 

V 

r 
~k 

-rp 

n'p 

~N~ 

-P 

-n 

* 

w 

-1 

"0 

no 

N 

0 

ri 

ni 

N 

+1 

rp 

np 

w 

+P 

M 

Plotting  y  against  ^  we  obtain  2  p  discrete  points  as  in  Fig.  3. 

When  N  is  large,  these  points,  are  somewhat  symmetrically  dis- 
tributed about  a  curve  of  the  general  form  illustrated  by  the 
dotted  line.  If  a  larger  number  of  observations  is  considered, 


ART.  29]     THE   LAW  OF   ACCIDENTAL   ERRORS 


33 


some  of  the  points  will  be  shifted  upward  while  others  will  be 
shifted  downward,  but  the  distribution  will  remain  approxi- 
mately symmetrical  with  respect  to  the  same  curve.  In  general, 
successive  equal  increments  to  N  cause  shifts  of  continually  de- 
creasing magnitude;  and  in  the  limit,  when  TV  becomes  equal  to 
infinity,  and  the  residuals  are  equal  to  the  accidental  errors,  the 
points  would  be  on  a  uniform  curve  symmetrical  to  the  yQ  ordi- 
nate.  The  curve  thus  determined  represents  the  relation  between 
the  magnitude  of  an  error  and  the  probability  of  its  occurrence 
in  a  given  series  of  measurements.  For  this  reason  it  is  called 
the  probability  curve. 


29.  Systems  of  Errors.  —  The  coordinates  of  the  probability 
curve  are  y  and-r-,  since  it  represents  the  distribution  of  the  true 

accidental  errors  AI,  A2,  etc.,  in  regard  to  relative  frequency  and 
magnitude.  Since  the  curve  is  uniform,  it  represents  not  only 
the  errors  of  the  actual  observations,  but  also  the  distribution  of 
all  of  the  accidental  errors  that  would  be  found  if  the  sensitive- 
ness of  our  instruments  were  infinitely  increased  and  an  infinite 
number  of  observations  were  made,  provided  only  that  all  of  the 
observations  were  made  with  the  same  degree  of  precision  and 
entirely  independently. 

All  of  the  errors  represented  by  a  curve  of  this  type  belong  to  a 
definite  system,  characterized  by  the  magnitude  of  the  maximum 
ordinate  yo  and  the  slope  of  the  curve.  Hence,  every  probability 
curve  represents  a  definite  system  of  errors.  It  also  represents 
the  accidental  errors  of  a  series  of  measurements  of  definite  pre- 
cision. Hence,  the  accidental  errors  of  series  of  measurements  of 
different  precision  belong  to  different  systems,  and  each  series 
is  characterized  by  a  definite  system  of  errors. 

The  probability  curves  A  and  B  in  Fig.  4  represent  the  systems 


34 


THE  THEORY  OF  MEASUREMENTS       [ART.  30 


of  errors  that  characterize  two  series  of  measurements  of  different 
precision.  As  the  precision  of  measurement  is  increased  it  is 
obvious  that  the  number  of  small  errors  will  increase  relatively 
to  the  number  of  large  ones.  Consequently  the  probability  of 
small  errors  will  be  greater  and  that  of  large  ones  will  be  less  in 
the  more  precise  series  A  than  in  the  less  precise  series  B.  Hence, 
the  curve  A  has  a  greater  maximum  ordinate  and  slopes  more 
rapidly  toward  the  horizontal  axis  than  the  curve  B. 


30.  The  Probability  Function.  —  The  maximum  ordinate  and 
the  slope  of  the  probability  curve  depend  on  the  constants  that 
appear  in  the  equation  of  the  curve.  When  we  know  the  form 
of  the  equation  and  have  a  method  of  determining  the  numerical 
value  of  the  constants,  we  are  able  to  determine  the  relative  pre- 
cision of  different  series  of  measurements.  Since  the  curve  repre- 
sents the  distribution  of  the  true  accidental  errors,  we  are  also  able 
to  compare  the  distribution  of  these  errors  with  that  of  the  resid- 
uals and  thus  develop  workable  methods  for  finding  the  most 
probable  numeric  of  the  measured  magnitude. 

It  is  obvious,  from  an  inspection  of  Figs.  3  and  4,  that  y  is  a 
continuous  function  of  A,  decreasing  very  rapidly  as  the  magni- 
tude of  A  increases  either  positively  or  negatively  and  symmetrical 
with  respect  to  the  y  axis.  Hence,  the  probability  curve  sug- 
gests an  equation  in  the  form 


(9) 


ART.  31]     THE  LAW  OF  ACCIDENTAL   ERRORS  35 

where  e  is  the  base  of  the  Napierian  system  of  logarithms,  o>  is  a 
constant  depending  on  the  precision  of  the  series  of  measurements 
considered,  and  the  other  variables  have  been  defined  above. 
This  equation  can  be  derived  analytically  from  the  three  axioms 
of  accidental  errors,  with  the  aid  of  several  plausible  assumptions 
regarding  the  constitution  of  such  errors,  or  from  the  principle 
of  the  arithmetical  mean.  However,  the  strongest  evidence  of 
its  exactness  lies  in  the  fact  that  it  gives  results  in  substantial 
agreement  with  experience.  Consequently,  we  will  adopt  it  as  an 
empirical  relation,  and  proceed  to  show  that  it  is  in  conformity 
with  the  three  axioms  and  leads  to  the  arithmetical  mean  as  the 
most  probable  numeric  derivable  from  a  series  of  equally  good 
independent  measurements  of  the  same  magnitude. 

Equation  (9)  is  the  mathematical  expression  of  the  law  of 
accidental  errors  and  is  often  referred  to  simply  as  the  law  of 
errors.  Its  right-hand  member  is  called  the  probability  function 
and,  for  the  sake  of  convenience,  is  represented  by  0  (A),  giving 
the  relations 

2/  =  0(A);    ^(A)^'™2^.  (10) 

31.  The  Precision  Constant.  —  The  curves  in  Fig.  4  were 
plotted,  to  the  same  scale,  from  data  computed  by  equation  (9). 
The  constant  w  was  taken  twice  as  great  for  the  curve  A  as  for 
the  curve  B,  and  in  both  cases  values  of  y  were  computed  for  suc- 
cessive integral  values  of  the  ratio  r--  The  maximum  ordinate  of 

each  of  these  curves  corresponds  to  the  zero  value  of  A  and  is 
equal  to  the  value  of  co  used  in  computing  the  y's.  The  curve 
A,  corresponding  to  the  larger  value  of  o>,  approaches  the  hori- 
zontal axis  much  more  rapidly  than  the  curve  B. 

Obviously,  the  constant  co  determines  both  the  maximum 
ordinate  and  the  slope  of  the  probability  curve.  But  we  have 
seen  that  these  characteristics  are  proportional  to  the  precision 
of  the  measurements  that  determine  the  system  of  errors  repre- 
sented. Hence  co  characterizes  the  system  of  errors  consid- 
ered and  is  proportional  to  the  precision  of  the  corresponding 
measurements.  Some  writers  have  called  it  the  precision  measure, 
but,  as  it  depends  only  on  the  accidental  errors  and  takes  no 
account  of  the  accuracy  with  which  constant  errors  are  avoided 
or  corrected,  it  does  not  give  a  complete  statement  of  the  pre- 


36  THE   THEORY  OF  MEASUREMENTS       [ART.  32 

cision.  Consequently  the  term  "  precision  measure  "  will  be  re- 
served for  a  function  to  be  discussed  later,  and  a;  will  be  called  the 
precision  constant  in  the  following  pages. 

When  A  is  taken  equal  to  zero  in  equation  (9),  y  is  equal  to  co. 
Hence  the  precision  of  measurements,  so  far  as  it  depends  upon 
accidental  errors,  is  proportional  to  the  probability  for  the  occur- 
rence of  zero  error  in  the  corresponding  system  of  errors.  In 
this  connection,  it  should  be  borne  in  mind  that  the  system  of 
errors  includes  all  of  the  errors  that  would  have  been  found 
with  an  infinite  number  of  observations,  and  that  it  cannot  be 
restricted  to  the  errors  of  the  actual  measurements  for  the  pur- 
pose of  computing  o>  directly.  Indirect  methods  for  computing 
a>  from  given  observations  will  be  discussed  later. 

32.  Discussion  of  the  Probability  Function.  —  Inspection  of 
the  curves  in  Fig.  4,  in  connection  with  equation  (9),  is  sufficient  to 
show  that  the  probability  function  is  in  agreement  with  the  first 
two  axioms.  Since  y  is  an  even  function  of  A,  positive  and  nega- 
tive errors  of  the  same  magnitude  are  equally  probable,  and  conse- 
quently equally  numerous  in  an  extended  series  of  measurements. 
Hence  the  first  axiom  is  fulfilled.  Since  A  enters  the  function 
only  in  the  negative  exponent,  the  probability  for  the  occurrence 
of  an  error  decreases  very  rapidly  as  its  magnitude  increases 
either  positively  or  negatively.  Hence  small  errors  are  much  more 
likely  to  occur  than  large  ones  and  the  second  axiom  is  fulfilled. 

Since  the  function  </>  (A)  is  continuous  for  values  of  A  ranging 
from  minus  infinity  to  plus  infinity,  it  is  apparently  at  variance 
with  the  third  axiom.  For,  if  all  of  the  errors  lie  between  definite 
finite  limits  —  L  and  +  L,  0  (A)  should  be  continuous  while  A 
lies  between  these  limits  and  equal  to  zero  for  all  values  of  A 
outside  of  them.  But  we  have  no  means  of  fixing  the  limits 
-f-  L  and  —  L,  in  any  given  case;  and  we  note  that  0(A)  becomes 
very  small  for  moderately  large  values  of  A.  Hence,  whatever  the 
true  value  of  L  may  be,  the  error  involved  in  extending  the  limits 
to  —oo  and  +00  is  infinitesimal.  Consequently,  </>(A)  is  in  sub- 
stantial agreement  with  the  third  axiom  provided  it  leads  to  the 
conclusion  that  all  possible  errors  lie  between  the  limits  —  oo  and 
+  oo .  This  will  be  the  case  if  it  gives  unity  for  the  probability 
that  a  single  error,  chosen  at  random,  lies  between  —  oo  and  -f  oo  . 
For,  if  all  of  the  errors  lie  between  these  limits,  the  probability 
considered  is  a  certainty  and  hence  is  represented  by  unity. 


ART. 33]      THE   LAW  OF   ACCIDENTAL   ERRORS 


37 


33.  The  Probability  Integral.  —  The  accidental  errors,  corre- 
sponding to  actual  measurements,  may  be  arranged  in  groups  ac- 
cording to  their  magnitude  in  the  same  manner  that  the  residuals 
were  arranged  in  article  twenty-eight.  When  this  is  done  the 
errors  in  succeeding  groups  differ  in  magnitude  by  an  amount 
equal  to  the  unit  error  kt  since  k  is  the  least  difference  that  can 
be  determined  with  the  instruments  used  in  making  the  obser- 
vations. Hence,  if  Ap  is  the  common  magnitude  of  the  errors 
in  the  pth  group, 


-A0  =  A 


(P+2) 


-A 


(p+i) 


or,  expressing  the  same  relation  in  different  form, 


where  a-  is  an  indeterminate  quantity  that  enters  each  of  the 
equations  because  we  do  not  know  the  actual  magnitude  of  the 

A's. 


FIG.  5. 

Let  the  probability  curve  in  Fig.  5  represent  the  system  of 
errors  to  which  the  errors  of  the  actual  measurements  belong. 
Then  the  ordinates  yp,  2/(p+i),  2/(P+2),  •  •  •  2/(p+a)  represent  the 
probabilities  of  the  errors  Ap,  A(p+i>,  .  .  .  A(p+e)  respectively. 
Since  the  errors  of  the  actual  measurements  satisfy  the  relation 
(i),  none  of  them  correspond  to  points  of  the  curve  lying  between 
the  ordinates  yp,  2/(P  +  i),  .  .  .  2/(P+«).  Hence,  in  virtue  of  equa- 
tion (6),  article  twenty-three,  if  we  choose  one  of  the  measure- 
ments at  random  the  probability  that  the  magnitude  of  its  error 
lies  between  Ap  and  A(P+Q)  is 


2/CP+8)- 


38  THE  THEORY  OF  MEASUREMENTS       [ART.  33 

Multiplying  and  dividing  the  second  member  by  q, 


where  ypq  is  written  for  the  mean  of  the  ordinates  between  yp 
and  2/(p+fl).     From  equation  (i) 


& 
Hence, 


In  the  limit,  when  we  consider  the  errors  of  an  infinite  number 
of  measurements  made  with  infinitely  sensitive  instruments,  every 
point  of  the  curve  represents  the  probability  of  one  of  the  errors 
of  the  system.  Consequently,  for  any  finite  value  of  q,  Ihe  inter- 
val between  the  ordinates  yp  and  y(P+q>  is  infinitesimal,  and  all 
of  the  ordinates  between  these  limits  may  be  considered  equal. 
Hence,  in  the  limit, 


p=       ,    ypq  =  2/A  = 
and  (iii)  reduces  to 


=*  (A)      ,  (11) 

where  y%+d*  represents  the  probability  that  the  magnitude  of  a 
single  error,  chosen  at  random,  is  between  A  and  A  +  dA. 

By  applying  the  usual  reasoning  of  the  integral  calculus,  it  is 
evident  that  the  expression 

rf  =  I  /%  (A)  «JA,  (12) 

/t  i/  a 

represents  the  probability  that  the  magnitude  of  an  error,  chosen 
at  random,  lies  between  the  limits  a  and  b.  The  integral  in  this 
expression  also  represents  the  area  under  the  probability  curve 

between  the  ordinates  at  T  and  T.     Consequently  the  probability 

in  question  is  represented  graphically  by  the  shaded  area  in  Fig.  6. 

The  probability  that  an  error,  chosen  at  random,  is  numerically 

less  than  a  given  error  A  is  equal  to  the  probability  that  it  lies 


ART.  33]     THE  LAW  OF  ACCIDENTAL  ERRORS  39 

between  the  limits  —A  and  -J-A.     Hence,  if  we  designate  this 
probability  by  PA, 


—  A 


—  A 


since  0  (A)  is  an  even  function  of  A.     Introducing  the  complete 
expression  for  0  (A)  from  equation  (10)  we  obtain 


A2 


k  jo 
For  the  sake  of  simplification,  put 

2A2 

then 


/Y'ett, 

Jo 


(13) 


which  is  an  entirely  general  expression  for  the  probability  PA, 
applicable  to  any  system  of  errors  when  we  know  the  correspond- 
ing values  of  the  constants  o>  and  k.  A  series  of  numerical  values 
of  the  right-hand  member  of  (13),  corresponding  to  successive 
values  of  the  argument  t,  is  given  in  Table  XI,  at  the  end  of 
this  volume.  Obviously,  this  table  may  be  used  in  computing 
the  probability  PA  corresponding  to  any  system  of  errors,  since 
the  characteristic  constants  o>  and  k  appear  only  in  the  limit  of  the 
integral. 

Whatever  the  values  of  the  constants  w  and  k,  the  limit  vVw  T 


40               THE   THEORY  OF  MEASUREMENTS  [ART.  34 

becomes  infinite  when  A  is  equal  to  infinity.     Hence,  in  every 
system  of  errors, 

*dt  =  l)  (13a) 


where  the  numerical  value  is  that  given  in  Table  XI,  for  the  limit 
t  equals  infinity.  Consequently  the  probability  function  0  (A) 
leads  to  the  conclusion  that  all  of  the  errors  in  any  system  lie 
between  the  limits  —  <x>  and  +00,  and,  therefore,  it  fulfills  the 
condition  imposed  by  the  third  axiom  as  explained  in  the  last 
paragraph  of  article  thirty-two. 

34.  Comparison  of  Theory  and  Experience.  —  Equation  (13) 
may  be  used  to  compare  the  distribution  of  the  residuals  actually 
found  in  any  series  of  measurements  with  the  theoretical  distri- 
bution of  the  accidental  errors.  If  N  equally  trustworthy  meas- 
urements of  the  same  magnitude  have  been  made,  all  of  the  N 
corresponding  accidental  errors  belong  to  the  same  system,  and 
the  probability  that  the  error  of  a  single  measurement  is  numer- 
ically less  than  A  is  given  by  PA  in  equation  (13).  Consequently, 
if  N  is  sufficiently  large,  we  should  expect  to  find 

#A  =  NP*  (iv) 

errors  less  than  A.  For,  if  we  consider  only  the  errors  of  the 
actual  measurements,  the  probability  that  one  of  them  is  less 
than  A  is  equal  to  the  ratio  of  the  number  less  than  A  to  the  total 
number.  In  the  same  manner,  the  number  less  than  A7  should 
be 


Hence,  the  number  lying  between  the  limits  A  and  A'  should  be 

N*  =  N*  -  N*.  (v) 

These  numbers  may  be  computed  by  equation  (13)  with  the  aid 
of  Table  XI,  when  we  know  N  and  the  value  of  the  expression 

V^co 

—  £—  corresponding  to  the  given  measurements.     The  number, 

Nrr  ,  of  residuals  lying  between  the  limits  r  equals  A  and  r'  equals 
A'  may  be  found  by  inspecting  the  series  of  residuals  computed 
from  the  given  measurements  by  equation  (3),  article  twenty-two. 
If  N  is  large  and  the  errors  of  the  given  measurements  satisfy 
the  theory  we  have  developed,  the  numbers  N%  and  Nrr'  should 


ART.  34]     THE   LAW  OF   ACCIDENTAL   ERRORS 


41 


be  very  nearly  equal,  since  in  an  extended  series  of  measurements 
the  residuals  are  very  nearly  equal  to  the  accidental  errors. 

The  following  illustration,  taken  from  Chauvenet's  "Manual 
of  Spherical  and  Practical  Astronomy,"  is  based  on  470  obser- 
vations of  the  right  ascension  of  Sirius  and  Altair,  by  Bradley. 
The  errors  of  these  measurements  belong  to  a  system  character- 
ized by  a  particular  value  of  the  ratio  T  that  has  been  computed, 

by  a  method  to  be  described  later  (articles  thirty-eight  and  forty- 
two),  and  gives  the  relation 

VTTCO 


k 


=  1.8086. 


Consequently,  to  find  the  theoretical  value  of  PA,  corresponding 
to  any  limit  A,  we  take  t  equal  to  1.8086  A  in  equation  (13)  and 
find  the  corresponding  value  of  the  integral  by  interpolation  from 
Table  XL 

The  third  column  of  the  following  table  gives  the  values  of 
PA  corresponding  to  the  chosen  values  of  A  in  the  first  column 
and  the  computed  values  of  t  in  the  second  column.  The  fourth 
column  gives  the  corresponding  values  of  N&.  computed  by  equa- 
tion (iv),  taking  N  equal  to  470.  The  sixth  column,  computed 
by  equation  (v),  gives  the  number,  Nj[,  of  errors  that  should 
lie  between  the  limits  A  and  A'  given  in  the  fifth.  The  seventh 
column  gives  the  number  of  residuals  actually  found  between  the 
same  limits. 


A 

t 

^A 

^A 

Limits 

A    A' 

< 

Nr 

// 

0.1 

0.1809 

0.2019 

95 

0.0-0.1 

95 

94 

0.2 

0.3617 

0.3910 

184 

0.1-0.2 

89 

88 

0.3 

0.5426 

0.5571 

262 

0.2-0.3 

78 

78 

0.4 

0.7234 

0.6937 

326 

0.3-0.4 

64 

58 

0.5 

0.9043 

0.7990 

376 

0.4-0.5 

50 

51 

0.6 

1.0852 

0.8751 

411 

0.5-0.6 

35 

36 

0.7 

1.2660 

0.9266 

436 

0.6-0.7 

"  25 

26 

0.8 

1.4469 

0.9593 

451 

0.7-0.8 

15 

14 

0.9 

1.6277 

0.9787 

460 

0.8-0.9 

9 

10 

1.0 

1.8086 

0.9895 

465 

0.9-1.0 

5 

7 

00 

GO 

1.0000 

470 

l.O-oo 

5 

8 

Comparison  of  the  numbers  in  the  last  two  columns  shows  very 
good  agreement  between  theory,  represented  by  N%,  and  expe- 


42  THE  THEORY  OF  MEASUREMENTS       [ART.  35 

rience,  represented  by  Nrrf,  when  we  remember  that  the  theory 
assumes  an  infinite  number  of  observations  and  that  the  series 
considered  is  finite.  Numerous  comparisons  of  this  nature  have 
been  made,  and  substantial  agreement  has  been  found  in  all 
cases  in  which  a  sufficient  number  of  independent  observations 
have  been  considered.  In  general,  the  differences  between  N% 
and  N^'  decrease  in  relative  magnitude  as  the  number  of  obser- 
vations is  increased. 

35.  The  Arithmetical  Mean.  —  In  article  twenty-four  it  was 
pointed  out,  as  one  of  the  fundamental  principles  of  the  theory 
of  errors,  that  the  arithmetical  mean  of  a  number  of  equally  trust- 
wor^hy  direct  measurements  on  the  same  magnitude  is  the  most 
probable  value  that  we  can  assign  to  the  numeric  of  the  measured 
magnitude.  In  order  to  show  that  the  probability  function  0  (A) 
leads  to  the  same  conclusion,  let  eft,  a2,  .  •  •  «AT  represent  the 
given  measurements,  and  let  x  represent  the  unknown  numeric 
of  the  measured  magnitude.  If  the  actual  value  of  this  numeric 
is  X,  the  true  accidental  errors  of  the  given  measurements  are 

Ai  =  ai  —  X,   A2  =  02  —  X,  .  .  .     AAT  =  ax  —  X,        (2) 

and  all  of  them  belong  to  the  same  system,  characterized  by  a 
particular  value  .of  the  precision  constant  co.  The  probability 
that  one  of  the  errors  of  this  system,  chosen  at  random,  is  equal 
to  an  arbitrary  magnitude  Ap  is  given  by  the  relation 


Since  we  cannot  determine  the  true  value  X,  the  most  probable 
value  that  we  can  assign  to  x  is  that  which  gives  a  maximum 
probability  that  N  errors  of  the  system  are  equal  to  the  N  resid- 
uals 

TI  =  ai  —  x,    rz  =  a2  —  x,     .  .  .  rN  =  aN  —  x.  (3) 

This  is  equivalent  to  determining  x,  so  that  the  residuals  are  as 
nearly  as  possible  equal  to  the  accidental  errors. 

If  2/1,  2/2,  ...  VN  represent  the  probabilities  that  a  single  error 
of  the  system,  chosen  at  random,  is  equal  to  r\,  r2,  .  .  .  rN  respec- 
tively, 

2/i  =  0  (n),     2/2  =  0  (r2),     .  .  .  yN  =  0 


Hence,  if  P  is  the  probability  that  N  of  the  errors  chosen  together 


ART.  35]     THE  LAW  OF   ACCIDENTAL   ERRORS  43 

are  equal  to  n,  r2,  .  .  .  rN  respectively,  we  have,  by  equation  (7), 
article  twenty-three, 

P  =  2/1  X  2/2  X  ...   X  yN 


Since  the  exponent  in  this  expression  is  negative  and  -^  is  con- 

K 

stant,  the  maximum  value  of  P  will  correspond  to  the  minimum 
value  of  (ri2  +  r22  +  .  .  .  -f  ?W2).  Hence  the  most  probable 
value  of  x  is  that  which  renders  the  sum  of  the  squares  of  the 
residuals  a  minimum. 

In  the  present  case,  the  r's  are  functions  of  a  single  independent 
variable  x.  Consequently  the  sum  of  the  squares  of  the  r's  will 
be  a  minimum  when  x  satisfies  the  condition 

-f-(ri2  +  r22  +  .  ...  +/VO  =0. 

(JJU 

Substituting  the  expression  for  the  r's  in  terms  of  x  from  equation 
(3)  this  becomes 


(a,  -  xY  +  (a2  -  xY  +  .  .  .  +  (a*  -  z)2    =  0. 
dx(  ) 

Hence,        («i  -  x)  +  (a2  -  x)  +  .  .  .  +  (aN  -  x)  =  0,  (14) 

ai  -f  «2  +  •  •  •  +  «AT 
and  x  =  —  jy- 

Consequently,  if  we  take  x  equal  to  the  arithmetical  mean  of  the 
a's  in  (3),  the  sum  of  the  squares  of  the  computed  r's  is  less  than 
for  any  other  value  of  x.  Hence  the  probability  P  that  N  errors 
of  the  system  are  equal  to  the  N  residuals  is  a  maximum,  and  the 
arithmetical  mean  is  the  most  probable  value  that  we  can  assign 
to  the  numeric  X  on  the  basis  of  the  given  measurements. 

Equation  (14)  shows  that  the  sum  of  the  residuals,  obtained 
by  subtracting  the  arithmetical  mean  from  each  of  the  given 
measurements,  is  equal  to  zero.  This  is  a  characteristic  property 
of  the  arithmetical  mean  and  serves  as  a  useful  check  on  the 
computation  of  the  residuals. 

The  argument  of  the  present  article  should  be  regarded  as  a 
justification  of  the  probability  function  0(A)  rather  than  as  a 
proof  of  the  principle  of  the  arithmetical  mean.  As  pointed  out 
above,  this  principle  is  sufficiently  established  on  a  priori  grounds 
and  by  common  consent. 


CHAPTER  V. 
CHARACTERISTIC   ERRORS. 

SEVERAL  different  derived  errors  have  been  used  as  a  measure 
of  the  relative  accuracy  of  different  series  of  measurements.  Such 
errors  are  called  characteristic  errors  of  the  system,  and  they  de- 
crease in  magnitude  as  the  accuracy  of  the  measurements,  on  which 
they  depend,  increases.  Those  most  commonly  employed  are  the 
average  error  A  ,  the  mean  error  M,  and  the  probable  error  E,  any 
one  of  which  may  be  used  as  a  measure  on  the  relative  accuracy 
of  a  single  observation. 

36.  The  Average  Error.  —  The  average  error  A  of  a  single 
observation  is  the  arithmetical  mean  of  all  of  the  individual  errors 
of  the  system  taken  without  regard  to  sign.  That  is,  all  of  the 
errors  are  taken  as  positive  in  forming  the  average.  Hence,  if 
N  is  the  total  number  of  errors, 


!  _ 

~N~  "W 

where  the  square  bracket  [  ]  is  used  as  a  sign  of  summation,  and 
the  ~~  over  the  A  indicates  that,  in  taking  the  sum,  all  of  the  A's 
are  to  be  considered  positive. 

In  accordance  with  the  usual  practice  of  writers  on  the  theory 
of  errors,  the  square  bracket  [  ]  will  be  used  as  a  sign  of  summa- 
tion, in  the  following  pages,  in  place  of  the  customary  sign  S. 
This  notation  is  adopted  because  it  saves  space  and  renders  com- 
plicated expressions  more  explicit. 

In  equation  (15)  all  of  the  errors  of  the  system  are  supposed 
to  be  included  in  the  summation.  Hence,  both  [A]  and  N  are 
infinite  and  the  equation  cannot  be  applied  to  find  A  directly 
from  the  errors  of  a  limited  number  of  measurements.  Conse- 
quently we  will  proceed  to  show  how  the  average  error  can  be 
derived  from  the  probability  function,  and  to  find  its  relation 
to  the  precision  constant  co.  A  little  later  we  shall  see  how  A 
can  be  computed  directly  from  the  residuals  corresponding  to  a 
limited  number  of  measurements. 

44 


ART.  36]  CHARACTERISTIC   ERRORS  45 

If  yd  is  the  probability  that  the  magnitude  of  a  single  error, 
chosen  at  random,  lies  between  A  and  A  +  dA,  and  rid  is  the  num- 
ber of  errors  between  these  limits, 


and  consequently 

nd  =  Nyd 

=  N4>  (A)  ^  (16) 

in  virtue  of  equation  (11),  article  thirty-three,  where  A  represents 
the  mean  magnitude  of  the  errors  lying  between  A  and  A  +  dA. 
Hence,  the  sum  of  the  errors  between  these  limits  is 


and  the  sum  of  the  errors  between  A  =  a  and  A  =  b  is 

N 


Substituting  the  complete  expression  for  </>(A)  from  equation  (10) 
this  becomes 


Hence,  the  sum  of  the  positive  errors  of  the  system  is 

Nu   /•»      -«*£, 
-;—  I     Ae        kz  dA, 
k    Jo 

and  the  sum  of  the  negative  errors  is 


Nu  r° 

k  J  -<* 


These  two  integrals  are  obviously  equal  in  magnitude  and  opposite 
in  sign.  Consequently  the  sum  of  all  of  the  errors  of  the  system 
taken  without  regard  to  sign  is 

Ae-^A  (17) 


7TCO 


46  THE  THEORY  OF  MEASUREMENTS       [ART.  37 

Hence  from  equation  (15), 


~  N 
and  introducing  the  numerical  value  of  IT, 

A  =0.3183--  (19) 

CO 

37.  The  Mean  Error.  —  The  mean  error  M  of  a  single  meas- 
urement in  a  given  series  is  the  square  root  of  the  mean  of  the 
squares  of  the  errors  in  the  system  determined  by  the  given 
measurements.  Expressed  mathematically 

A^  +  A^-f-.*  +  A^_[A1 
N  '   N 

This  equation  includes  all  of  the  errors  that  belong  to  the  given 
system.     Hence,  as  pointed  out  in  article  thirty-six,  in  regard  to 
equation  (15),  it  cannot  be  applied  directly  to  a  limited  series  of 
measurements. 
By  equation  (16)  the  number  of  errors  with  magnitudes  between 

the  limits  A  and  A  +  dA  is  equal  to  ,  —  .     Consequently 

/c 

the  sum  of  the  squares  of  the  errors  between  these  limits  is  equal 
#A24>(A)dA 

k 
in  the  last  article, 


to  -        .;    .     Hence,  by  reasoning  similar  to  that  employed 


(21) 

/»« 

/ 


2N«  r»A%-,  * 


since  the  integrand  is  an  even  function  of  A.     Integrating  by 
parts, 


7TCO 


The  first  term  of  the  second  member  of  this  equation  reduces  to 


AKT.38]  CHARACTERISTIC   ERRORS  47 

zero  when  the  limits  are  applied.     Putting  t2  for  in  the 

K 

second  term, 

[Al-^P^a-™  (22) 

TT^CO2  Jo  2  7TC02 

in  virtue  of  equation  (13a).     Hence, 


N       2™* 
and 

M  = 


=  0.3989-- 

CO 


(23) 


38.  The  Probable  Error.  —  The  probable  error  E  of  a  single 
measurement  is  a  magnitude  such  that  a  single  error,  chosen  at 
random  from  the  given  system,  is  as  likely  to  be  numerically 
greater  than  E  as  less  than  E.  In  other  words,  the  probability 
that  the  error  of  a  single  measurement  is  greater  than  E  is  equal 
to  the  probability  that  it  is  less  than  E.  Hence,  in  any  extended 
series  of  measurements,  one-half  of  the  errors  are  less  than  E  and 
one-half  of  them  are  greater  than  E. 

The  name  "  probable  error,"  though  sanctioned  by  universal 
usage,  is  unfortunate;  and  the  student  cannot  be  too  strongly 
cautioned  against  a  common  misinterpretation  of  its  meaning. 
The  probable  error  is  NOT  the  most  probable  magnitude  of  the 
error  of  a  single  measurement  and  it  DOES  NOT  determine  the 
limits  within  which  the  true  numeric  of  the  measured  magnitude 
may  be  expected  to  lie.  Thus,  if  x  represents  the  measured 
numeric  of  a  given  magnitude  Q  and  E  is  the  probable  error  of  x, 
it  is  customary  to  express  the  result  of  the  measurement  in  the 
form 

Q  =  x  ±  E. 

This  does  not  signify  that  the  true  numeric  of  Q  lies  between  the 
limits  x  —  E  and  x  +  E,  neither  does  it  imply  that  x  is  probably 
in  error  by  the  amount  E.  It  means  that  the  numeric  of  Q  is  as 
likely  to  lie  between  the  above  limits  as  outside  of  them.  If  a 
new  measurement  is  made  "by  the  same  method  and  with  equal 
care,  the  probability  that  it  will  differ  from  x  by  less  than  E  is 
equal  to  the  probability  that  it  will  differ  by  more  than  E. 


48 


THE  THEORY  OF  MEASUREMENTS       [ART.  38 


In  article  thirty-three  it  was  pointed  out  that  the  probability 
that  an  error,  chosen  at  random  from  a  given  system,  lies  between 
the  limits  A  =  a  and  A  =  b  is  represented  by  the  area  under  the 
probability  curve  between  the  ordinates  corresponding  to  the 
limiting  values  of  A.  Hence,  the  probability  that  the  error  of  a 
single  measurement  is  numerically  less  than  E  may  be  represented 
by  the  area  under  the  probability  curve  between  the  ordinates  y-E 
and  y+E,  in  Fig.  7,  and  the  probability  that  it  is  greater  than  E  by 
the  sum  of  the  areas  outside  of  these  ordinates.  Since  these  two 


FIG.  7. 

probabilities  are  equal,  by  definition,  the  ordinates  correspond- 
ing to  the  probable  error  bisect  the  areas  under  the  two  branches 
of  the  probability  curve. 

Since  the  probability  that  the  error  of  a  single  measurement  is 
less  than  E  is  equal  to  the  probability  that  it  is  greater  than  E 
and  the  probability  that  it  is  less  than  infinity  is  unity,  the 
probability  that  it  is  less  than  E  is  one-half.  Consequently, 
putting  A  equal  to  E  in  equation  (13),  article  thirty-three, 


Pw  =  ~ 


rw  T"  1 

e-«dt  -  2- 

\J 

From  Table  XI, 

PA  =  0.49375  for  the  limit  t  =  0.47, 
PA  =  0.50275  for  the  limit  t  =  0.48, 

and  by  interpolation, 

PE  =  0.50000  for  the  limit  t  =  0.47694. 
Hence,  equation  (24)  is  satisfied  when 


(24) 


=  0.47694, 


ART.  39] 
and  we  have 


CHARACTERISTIC   ERRORS 


E 


0.47694    k 

VTT        w 


=  0.2691  - 

CO 


49 


(25) 


39.    Relations  between  the  Characteristic  Errors.  —  Elimina- 

k 
ting-  from  equations  (18),  (23),  and  (25),  taken  two  at  a  time,  we 

obtain  the  relations 


(26") 
E  =  0.4769  •  VTT  -A  =  0.8453  -A, 

E  =  0.4769  •  V2  •  M  =  0.6745  •  M,. 

which  express  the  relative  magnitudes  of  the  average,  mean,  and 
probable  errors.     These  relations  are  universally  adopted  in  com- 


MAE 
k  k  k 


FIG.  8. 


puting  the  precision  of  given  series  of  measurements,  and  they 
should  be  firmly  fixed  in  mind. 

The  three  equations  from  which  the  relations  (26)  are  derived 
may  be  put  in  the  form 

A  =  0.3183 

k    co 
M  _  0.3989 
k     co 
E  =  0.2691 

k    co 

The  probability  curve  in   Fig.  8  represents  the   distribution  of 
the  errors  in  a  system  characterized  by  a  particular  value  of  co, 


(27) 


50  THE  THEORY  OF  MEASUREMENTS       [ART.  39 

determined  by  a  given  series  of  measurements.     The  ordinates 

AM       AE 
VA>  VM>  and  Us  correspond  to  the  abscissae  -^>  -jp  and-"&  >  com" 

puted  by  the  above  equations.  Consequently,  yA  represents  the 
probability  that  the  error  of  a  single  measurement  is  equal  to 
+A,  yM  the  probability  that  it  is  equal  to  +M,  and  yE  the  prob- 
ability that  it  is  equal  to  +E.  In  like  manner  y-A,  y-M,  and 
y~E  represent  the  respective  probabilities  for  the  occurrence  of 
errors  equal  to  —A,  —M,  and  —E. 

A  curve  of  this  type  can  be  constructed  to  correspond  to  any 
given  series  of  measurements,  and  in  all  cases  the  relative  loca- 
tion of  the  ordinates  yA,  yM)  and  yE  will  be  the  same.  It  was 
pointed  out  in  the  last  article  that  the  ordinates  yE  and  y-E  bisect 
the  areas  under  the  two  branches  of  the  curve.  Consequently, 
in  an  extended  series  of  measurements,  somewhat  more  than  one- 
half  of  the  errors  will  be  less  than  either  the  average  or  the  mean 
error.  Moreover,  it  is  obvious  from  Fig.  8  that  an  error  equal  to 
E  is  somewhat  more  likely  to  occur  than  one  equal  to  either  A  or  M. 

Since  each  of  the  characteristic  errors  A,  M,  and  E,  bears  a 
constant  relation  to  the  precision  constant  co,  any  one  of  them 
might  be  used  as  a  measure  of  the  precision  of  a  single  measure- 
ment in  a  given  series,  so  far  as  this  depends  on  accidental  errors. 
The  probable  error  is  more  commonly  employed  for  this  purpose 
on  account  of  its  median  position  in  the  system  of  errors  deter- 
mined by  the  given  measurements. 

It  is  interesting  to  observe  that  the  ordinate  yM  corresponds  to 
a  point  of  inflection  in  the  probability  curve.  By  the  ordinary 
method  of  the  calculus  we  know  that  this  curve  has  a  point  of 
inflection  corresponding  to  the  abscissa  that  satisfies  the  relation 


Substituting  the  complete  expression  for  y 


Hence, 


ART.  40]  CHARACTERISTIC  ERRORS  51 

is  the  abscissa  of  the  point  of  inflection.     Comparing  this  with 
equation  (23)  we  see  that 


and  consequently  that  the  ordinates  yM  and  y-M  meet  the  prob- 
ability curve  at  points  of  inflection. 

40.   Characteristic  Errors  of  the  Arithmetical  Mean.  —  Equa- 
tion (23)  may  be  put  in  the  form 

CO2  1 


where  M  is  the  mean  error  of  a  single  measurement  in  a  series 
corresponding  to  the  unit  error  k  and  the  precision  constant  w. 
Consequently  the  probability  function, 

"***& 

y  =  we       k  y 
corresponding  to  the  same  series  may  be  put  in  the  form 

y  =  ae    2M*.  (i) 

If  A  i,  A  2,  .  .  .  AJV  are  the  accidental  errors  of  N  direct  measure- 
ments in  the  same  series,  the  probability  P  that  they  all  occur  in 
a  system  characterized  by  the  mean  error  M  is  equal  to  the  product 
of  the  probabilities  for  the  occurrence  of  the  individual  errors  in 
that  system.  Hence, 


If   the   individual   measurements   are   represented   by  a\t   0,2, 
.  .  .  aN,  and  the  true  numeric  of  the  measured  quantity  is  X, 

Ai  =  ai  -  X;    A2  =  az  -  X\     .  .  .  A#  =  aN  -  X, 

and,  if  x  is  the  arithmetical  mean  of  the  measurements,  the  corre- 
sponding residuals  are 

n  =  ai  —  x',    rz  =  «2  —  x;    .  .  .  rN  =  aN  —  x. 
Consequently,  if  the  error  of  the  arithmetical  mean  is  5, 

X  -  x  =  5, 
and 

Ai  =  n  -  5;     A2  =  r2  -  5;     .  .  .  A#  =  rN  —  8. 

Squaring  and  adding, 

[A2]  =  [r2]-25M+ATS2; 

(28) 


52  THE  THEORY  OF  MEASUREMENTS       [ART.  40 

since  [r]  Is  equal  to  zero  in  virtue  of  equation  (14),  article  thirty- 
five.  When  this  value  of  [A2]  is  substituted  in  (ii),  the  resulting 
value  of  P  is  the  probability  that  the  arithmetical  mean  is  in 
error  by  an  amount  6.  For,  as  we  have  seen  in  article  thirty-five, 
the  minimum  value  of  [r2]  occurs  when  x  is  taken  equal  to  the 
arithmetical  mean.  Consequently,  P  is  a  maximum  when  <5  is 
equal  to  zero  and  decreases  in  accordance  with  the  probability 
function  as  5  increases  either  positively  or  negatively. 

We  do  not  know  the  exact  value  of  either  X  or  5;  but,  if  ya  is 
the  probability  that  the  error  of  the  arithmetical  mean  is  equal 
to  an  arbitrary  magnitude  5,  the  foregoing  reasoning  leads  to  the 
relation 


2M2 


But  the  arithmetical  mean  is  equivalent  to  a  single  measurement 
in  a  series  of  much  greater  precision  than  that  of  the  given  meas- 
urements. Hence,  if  o>a  is  the  precision  constant  correspond- 
ing to  this  hypothetical  series  and  Ma  is  the  mean  error  of  the 
arithmetical  mean,  we  have  by  analogy  with  (i) 

a* 

ya  =  wae    2  M«2  .  (iv) 

Equations  (iii)  and  (iv)  are  two  expressions  for  the  same  prob- 
ability and  should  give  equal  values  to  ya  whatever  the  assumed 
value  of  5.  This  is  possible  only  when 


2M, 


and 

1  N 


~  2M2 
Hence, 

M          M 

Ma  =  —  =•• 

VN 

Consequently,  the  mean%  error  of  the  arithmetical  mean  is  equal 
to  the  mean  error  of  a  single  measurement  divided  by  the  square 
root  of  the  number  of  measurements. 

Since  the  average,  mean,  and  probable  errors  of  a  single  meas- 
urement are  connected  by  the  relations  (26),  the  corresponding 


Art.  41]  CHARACTERISTIC   ERRORS  53 

errors  of  the  arithmetical  mean,  distinguished  by  th.e  subscript 
a,  are  given  by  the  relations 

40  =  -4=;     Ma  =  -^=;      Ea  =  -?j=.  (29) 

VN  VN  VN 

41.   Practical    Computation    of    Characteristic*  Errors.  —  As 

pointed  out  in  article  thirty-seven,  the  square  of  the  mean  error 

[A21 
M  is  the  limiting  value  of  the  ratio  ^rp  when  both  members 

become  infinite,  i.e.,  when  all  of  the  errors  of  the  given  system 
are  considered.  But  the  errors  of  the  actual  measurements  fall 
into  groups,  as  explained  in  article  thirty-three,  and  the  errors  in 
succeeding  groups  differ  in  magnitude  by  a  constant  amount  k, 
depending  on  the  nature  of  the  instruments  used  in  making  the 
observations.  Consequently,  the  ordinates,  of  the  probability 
curve,  corresponding  to  these  errors  are  uniformly  distributed 
along  the  horizontal  axis.  Hence,  if  we  include  in  [A2]  only  the 
errors  of  the  actual  measurements,  the  limiting  value  of  the  ratio 

fA2l 

L-^-  when  N  is  indefinitely  increased  will  be  nearly  the  same  as  if 

all  of  the  errors  of  the  system  were  included.  Since  the  ratio 
approaches  its  limit  very  rapidly  as  N  increases,  the  value  of  M 
can  be  determined,  with  sufficient  precision  for  most  practical 
purposes,  from  a  somewhat  limited  series  of  measurements. 

If  we  knew  the  true  accidental  errors,  the  mean  error  could  be 
computed  at  once  from  the  relation 

(v) 

and,  since  the  residuals  are  nearly  equal  to  the  accidental  errors 
when  N  is  very  large,  an  approximate  value  can  be  obtained  by 
using  the  r's  in  place  of  the  A's.  A  better  approximation  can  be 
obtained  if  we  take  account  of  the  difference  between  the  A's 
and  the  r's.  From  equation  (28) 

[A2]  =  [r2]  +  AT52,  (vi) 

where  6  is  the  unknown  error  of  the  arithmetical  mean.  Probably 
the  best  approximation  we  can  make  to  the  true  value  of  8  is  to 
set  it  equal  to  the  mean  error  of  the  arithmetical  mean.  Hence, 
from  the  second  of  equations  (29) 


54  THE  THEORY  OF  MEASUREMENTS       [ART.  41 


Consequently,  (vi)  becomes 

NM2  =  [r2]  + 
and  we  have 

(30) 

Thus  the  square  of  the  mean  error  of  a  single  measurement  is 
equal  to  the  sum  of  the  squares  of  the  residuals  divided  by  the 
number  of  measurements  less  one. 

Combining  (30)  with  the  third  of  equations  (26),  article  thirty- 
nine,  we  obtain  the  expression 


E  =  0.6745  V^rj  <31) 

for  the  probable  error  of  a  single  measurement.  Hence,  by  equa- 
tions (29),  the  mean  error  Ma  and  the  probable  error  Ea  of  the 
arithmetical  mean  are  given  by  the  relations 


and  *°  =  °-  (32) 


When  the  number  of  measurements  is  large,  the  computation 
of  the  probable  errors  E  and  Ea  by  the  above  formulae  is  some- 
what tedious,  owing  to  the  necessity  of  finding  the"  square  of 
each  of  the  residuals.  In  such  cases  a  sufficiently  close  approx- 
imation for  practical  purposes  can  be  derived  from  the  average 
error  A  with  the  aid  of  equations  (26).  The  first  of  these  equa- 
tions may  be  written  in  the  form 

[A3  =  T  [A]2 

N       2  N2' 

If  we  assume  that  the  distribution  of  the  residuals  is  the  same  as 
that  of  the  true  accidental  errors,  a  condition  that  is  accurately 
fulfilled  when  N  is  very  large,  we  can  put 


N 
Consequently, 


ART.  41]  CHARACTERISTIC   ERRORS  55 

When  the  mean  error  M  is  expressed  in  terms  of  the  A's,  equation 
(30)  becomes 

[A2]_     M 
N  '    N-l' 
or 

[Ag  =    N       [Sp. 

[r2]       tf-  1       [r]2  ' 
Consequently 

[A?  [r? 


and,  since  this  ratio  is  equal  to  A2,  we  have 


==     and     A0  =  -  X  (33) 

-1)  NVN-1 

Combining  this  result  with  the  second  of  equations  (26)  and  the 
third  of  (29),  we  obtain 

E  =  0.8453     .     ^         ;  Ea  =  0.8453  -  ^        .       (34) 

VN(N-1)'  NVN-1 

The  above  formulae  for  computing  the  characteristic  errors  from 
the  residuals  have  been  derived  on  the  assumption  that  the  true 
accidental  errors  and  the  residuals  follow  the  same  law  of  dis- 
tribution. This  is  strictly  true  only  when  the  number  of  measure- 
ments considered  is  very  large.  Yet,  for  lack  of  a  better  method, 
it  is  customary  to  apply  the  foregoing  formulas  to  the  discussion 
of  the  errors  of  limited  series  of  measurements  and  the  results 
thus  obtained  are  sufficiently  accurate  for  most  practical  purposes. 
When  the  highest  attainable  precision  is  sought,  the  number  of 
observations  must  be  increased  to  such  an  extent  that  the  theo- 
retical conditions  are  fulfilled. 

The  choice  between  the  formulae  involving  the  average  error 
A  and  those  depending  on  the  mean  error  M  is  determined  largely 
by  the  number  of  measurements  available  and  the  amount  of 
time  that  it  is  worth  while  to  devote  to  the  computations.  When 
the  number  of  measurements  is  very  large,  both  sets  of  formulae 
lead  to  the  same  values  for  the  probable  errors  E  and  Ea,  and 
much  time  is  saved  by  employing  those  depending  on  A.  For 
limited  series  of  observations  a  better  approximation  to  the  true 
values  of  these  errors  is  obtained  by  employing  the  formulae  in- 
volving the  mean  error.  In  either  case  the  computation  may  be 


56 


THE   THEORY  OF  MEASUREMENTS       [ART.  42 


facilitated  by  the  use  of  Tables  XIV  and  XV  at  the  end  of  this 
volume.     These  tables  give  the  values  of  the  functions 

0.6745  0.8453  0.8453 


0.6745 


VN(N-1)' 


and 


NVN-l' 


corresponding  to  all  integral  values  of  N  between  two  and  one 
hundred. 

42.  Numerical  Example.  —  The  following  example,  represent- 
ing a  series  of  observations  taken  for  the  purpose  of  calibrating 
the  screw  of  a  micrometer  microscope,  will  serve  to  illustrate  the 
practical  application  of  the  foregoing  methods.  Twenty  inde- 
pendent measurements  of  the  normal  -distance  between  two 
parallel  lines,  expressed  in  terms  of  the  divisions  of  the  micrometer 
head,  are  given  in  the  first  and  fourth  columns  of  the  following 
table  under  a. 


a 

r 

ri 

a 

r 

r2 

194.7 

+0.53 

0.2809 

194.3 

+0.13 

0.0169 

194.1 

-0.07 

0.0049 

194.3 

+0.13 

0.0169 

194.3 

+0.13 

0.0169 

194.0 

-0.17 

0.0289 

194.0 

-0.17 

0.0289 

194.4 

+0.23 

0.0529 

193.7 

-0.47 

0.2209 

194.5 

+0.33 

0.1089 

194.1           -0.07 

0.0049 

193.8 

-0.37 

0.1369 

193.9           -0.27 

0.0729 

193.9 

-0.27 

0.0729 

194.3           +0.13 

0.0169 

193.9 

-0.27 

0.0729 

194.3           +0.13 

0.0169 

194.8 

+0.63 

0.3969 

194.4           +0.23 

0.0529 

193.7 

-0.47 

0.2209 

194.17 

5.20 

1.8420 

.r 

0 

[r2] 

Since  the  observations  are  independent  and  equally  trust- 
worthy, the  most  probable  value  that  we  can  assign  to  the  numeric 
of  the  measured  magnitude  is  the  arithmetical  mean  x;  and  we 
find  that  x  is  equal  to  194.17  micrometer  divisions.  Subtracting 
194.17  from  each  of  the  given  observations  we  obtain  the  residuals 
in  the  columns  under  r.  The  algebraic  sum  of  these  residuals  is 
equal  to  zero  as  it  should  be,  owing  to  the  properties  of  the  arith- 
metical mean.  The  sum  without  regard  to  sign,  [r],  is  equal  to 
5.20.  Squaring  each  of  the  residuals  gives  the  numbers  in  the 
columns  under  r2  and  adding  these  figures  gives  1.8920  for  the 
sum  of  the  squares  of  the  residual  [r2]. 

Taking  N  equal  to  twenty,  in  formulae  (33)  and  (34),  we  find 
the  average  and  probable  errors 


ART.  42]  CHARACTERISTIC   ERRORS  57 

=  =b  0.267;    Aa  =  Ar    ^          =  ±  0.0596, 

NVN-l 

E  =  0.8453—  7==  =  ±0.226;  #«  =  0.8453  ^-^  =  =  ±0.0504, 


where  the  numerical  results  are  written  with  the  indefinite  sign  ± 
since  the  corresponding  errors  are  as  likely  to  be  positive  as  nega- 
tive. 

When  formulae  (30),  (31),  and  (32)  are  employed  we  obtain  the 
mean  errors, 


and  the  probable  errors 

E  =  0.6745 


The  values  of  the  probable  errors  E  and  jEk,  computed  by  the 
two  methods,  agree  as  closely  as  could  be  expected  with  so  small 
a  number  of  observations.  Probably  the  values  d=  0.210  and 
±  0.047,  computed  from  the  mean  errors  M  and  Ma,  are  the  more 
accurate,  but  those  derived  from  the  average  errors  A  and  Aa  are 
sufficiently  exact  for  most  practical  purposes.  An  inspection  of 
the  column  of  residuals  is  sufficient  to  show  that  eleven  of  them 
are  numerically  greater,  and  nine  are  numerically  less  than  either 
of  the  computed  values  of  E.  Consequently,  both  of  these  values 
fulfill  the  fundamental  definition  of  the  probable  error  of  a  single 
measurement  as  nearly  as  we  ought  to  expect  when  only  twenty 
observations  are  considered. 

If  we  use  D  to  represent  the  measured  distance  between  the 
parallel  lines,  in  terms  of  micrometer  divisions,  we  may  write 
the  final  result  of  the  measurements  in  the  form 

D  =  194.170  =t  0.047  mic.  div. 

This  does  not  mean  that  the  true  value  of  D  lies  between  the 
specified  limits,  but  that  it  is  equally  likely  to  lie  between  these 
limits  or  outside  of  them.  Thus,  if  another  and  independent 
series  of  twenty  measurements  of  the  same  distance  were  made 


58  THE  THEORY  OF  MEASUREMENTS       [ART.  43 

with  the  same  instrument,  and  with  equal  care,  the  chance  that 
the  final  result  would  lie  between  194.123  and  194.217  is  equal  to 
the  chance  that  it  would  lie  outside  of  these  limits. 

Equation  (25),  article  thirty-eight,  may  be  written  in  the  form 

-co      0.4769 


Taking  E  equal  to  0.210,  we  find  that 

v££  =  2.271 
k 

for  the  particular  system  of  errors  determined  by  the  above  meas- 
urements. Consequently,  the  probability  for  the  occurrence  of  an 
error  less  than  A  in  this  system  is,  by  equation  (13),  article  thirty- 
three, 

»2.271.A 


and,  since  there  are  twenty  measurements,  we  should  expect  to 
find  20  PA  errors  numerically  less  than  any  assigned  value  of  A. 

The  values  of  PA,  corresponding  to  various  assigned  values  of 
A,  can  be  easily  computed  with  the  aid  of  Table  XI  and  applied, 
as  explained  in  article  thirty-four,  to  compare  the  theoretical 
distribution  of  the  accidental  errors  with  that  of  the  residuals 
given  under  r  in  the  above  table.  Such  a  comparison  would  have 
very  little  significance  in  the  present  case,  however  it  resulted, 
since  the  number  of  observations  considered  is  far  too  small  to 
fulfill  the  theoretical  requirements.  But  it  would  show  that, 
even  in  such  extreme  cases,  the  deviations  from  the  law  of  errors 
are  not  greater  than  might  be  expected.  The  actual  comparison 
is  left  as  an  exercise  for  the  student. 

43.  Rules  for  the  Use  of  Significant  Figures.  —  The  funda- 
mental principles  underlying  the  use  of  significant  figures  were 
explained  in  article  fifteen.  General  rules  for  their  practical  ap- 
plication may  be  stated  in  terms  of  the  probable  error  as  follows: 

All  measured  quantities  should  be  so  expressed  that  the  last 
recorded  significant  figure  occupies  the  place  corresponding  to  the 
second  significant  figure  in  the  probable  error  of  the  quantity 
considered. 

The  number  of  significant  figures  carried  through  the  compu- 


ART.  43]  CHARACTERISTIC  ERRORS  59 

tations  should  be  sufficient  to  give  the  final  result  within  one  unit 
in  the  last  place  retained  and  no  more. 

For  practical  purposes  probable  errors  should  be  computed  to 
two  significant  figures. 

The  example  given  in  the  preceding  article  will  serve  to  illus- 
trate the  application  of  these  rules.  The  second  significant  figure 
in  the  probable  error  of  the  arithmetical  mean  occupies  the  third 
decimal  place.  Consequently,  the  final  result  is  carried  to  three 
decimal  places,  notwithstanding  the  fact  that  the  last  place  is 
occupied  by  a  zero.  It  would  obviously  be  useless  to  carry  out 
the  result  farther  than  this,  since  the  probable  error  shows  that 
the  digit  in  the  second  decimal  place  is  equally  likely  to  be  in 
error  by  more  or  less  than  .five  units.  If  less  significant  figures 
were  used,  the  fifth  figure  in  computed  results  might  be  vitiated 
by  more  than  one  unit. 

In  order  to  apply  the  rules  to  the  individual  measurements,  it 
is  necessary  to  make  a  preliminary  series  of  observations,  under 
as  nearly  as  possible  the  same  conditions  that  will  prevail  during 
the  final  measurements,  and  compute  the  probable  error  of  a 
single  observation  from  the  data  thus  obtained.  Then,  if  possible, 
all  final  measurements  should  be  recorded  to  the  second  significant 
figure  in  this  probable  error  and  no  farther.  It  sometimes  happens, 
as  in  the  above  example,  that  the  graduation  of  the  measuring 
instruments  used  is  not  sufficiently  fine  to  permit  the  attainment 
of  the  number  of  significant  figures  required  by  the  rule.  In  such 
cases  the  observations  are  recorded  to  the  last  attainable  figure, 
.or,  if  possible,  the  instruments  are  so  modified  that  they  give 
the  required  number  of  figures.  Thus,  in  the  example  cited,  the 
second  significant  figure  in  the  probable  error  of  a  single  measure- 
ment is  in  the  second  decimal  place,  but  the  micrometer  can 
be  read  only  to  one-tenth  of  a  division.  Hence  the  individual 
measurements  are  recorded  to  the  first  instead  of  the  second 
decimal  place.  In  this  case  the  accuracy  attained  in  making  the 
settings  of  the  instrument  was  greater  than  that  attained  in 
making  the  readings,  and  an  observer,  with  sufficient  experience, 
would  be  justified  in  estimating  the  fractional  parts  to  the  nearest 
hundredth  of  a  division.  A  better  plan  would  be  to  provide  the 
micrometer  head  with  a  vernier  reading  to  tenths  or  hundredths  of 
a  division.  In  the  opposite  case,  when  the  accuracy  of  setting  is 
less  than  the  attainable  accuracy  of  reading,  it  is  useless  to  record 


60  THE   THEORY  OF  MEASUREMENTS       [ART.  43 

the  readings  beyond  the  second  significant  figure  in  the  probable 
error  of  a  single  observation. 

For  the  purpose  of  computing  the  residuals,  the  arithmetical 
mean  should  be  rounded  to  such  an  extent  that  the  majority  of 
the  residuals  will  come  out  with  two  significant  figures.  This 
greatly  reduces  the  labor  of  the  computations  and  gives  the  calcu- 
lated characteristic  errors  within  one  unit  in  the  second  significant 
figure. 


CHAPTER  VI. 
MEASUREMENTS   OF  UNEQUAL  PRECISION. 

44.  Weights  of  Measurements.  —  In  the  preceding  chapter 
we  have  been  dealing  with  measurements  of  equal  precision,  and 
the  results  obtained  have  been  derived  on  the  supposition  that 
there  was  no  reason  to  assume  that  any  one  of  the  observations 
was  better  than  any  other.  Under  these  conditions  we  have 
seen  that  the  most  probable  value  that  we  can  assign  to  the 
numeric  of  the  measured  magnitude  is  the  arithmetical  mean  of 
the  individual  observations.  Also,  if  M  and  E  are  the  mean  and 
probable  errors  of  a  single  observation,  Ma  and  Ea  the  mean  and 
probable  errors  of  the  arithmetical  mean,  and  A/"  the  number  of 
observations,  we  have  the  relations 

#  =  0.6745  M;    ' Ea  =  0.6745  Mn, 
M  E 


v 


(35) 


The  true  numeric  X  of  the  measured  magnitude  cannot  be 
exactly  determined  from  the  given  observations,  but  the  final 
result  of  the  measurements  may  be  expressed  in  the  form 

X  =  x  ±  Ea, 

which  signifies  that  X  is  as  likely  to  lie  between  the  specified 
limits  as  outside  of  them. 

Now  suppose  that  the  results  of  m  independent  series  of  meas- 
urements of  the  same  magnitude,  made  by  the  same  or  different 
methods,  are  given  in  the  form 

X  =  xi±  Elt 
X  =  x%  it  EZ, 

X  =  xm  d=  Em. 
61 


62  THE   THEORY  OF  MEASUREMENTS       [ART.  44 

What  is  the  most  probable  value  that  can  be  assigned  to  X  on 
the  basis  of  these  results?  Obviously,  the  arithmetical  mean  of  the 
x's  will  not  do  in  this  case,  unless  the  E's  are  all  equal,  since  the 
x's  violate  the  condition  on  which  the  principle  of  the  arithmetical 
mean  is  founded.  If  we  knew  the  individual  observations  from 
which  each  of  the  x's  were  derived,  and  if  the  probable  error  of 
a  single  observation  was  the  same  in  each  of  the  series,  the  most 
probable  value  of  X  would  be  given  by  the  arithmetical  mean  of 
all  of  the  individual  observations.  Generally  we  do  not  have  the 
original  observations,  and,  when  we  do,  it  frequently  happens  that 
the  probable  error  of  a  single  observation  is  different  in  the  differ- 
ent series.  Consequently  the  direct  method  is  seldom  applicable. 

The  E's  may  differ  on  account  of  differences  in  the  number  of 
observations  in  the  several  series,  or  from  the  fact  that  the  prob- 
able error  of  a  single  observation  is  not  the  same  in  all  of  them,  or 
from  both  of  these  causes.  Whatever  the  cause  of  the  difference, 
it  is  generally  necessary  to  reduce  the  given  results  to  a  series  of 
equivalent  observations  having  the  same  probable  error  before 
taking  the  mean.  For  it  is  obvious  that  a  result  showing  a  small 
probable  error  should  count  for  more,  or  have  greater  weight, 
in  determining  the  value  of  X  than  one-  that  corresponds  to  a 
large  probable  error,  since  the  former  result  has  cost  more  in  time 
and  labor  than  the  latter. 

The  reduction  to  equivalent  observations  having  the  same 
probable  error  is  accomplished  as  follows:  m  numerical  quanti- 
ties wi,  w2,  .  .  .  wm,  called  the  weights  of  the  quantities  Xi,  x2, 
.  .  .  xm,  are  determined  by  the  relations 

E*  Ea2  E* 

W^E?>    W*=Ef'>    '••  'Wm=E^'  (36) 

where  Ea  is  an  arbitrary  quantity,  generally  so  chosen  that  all 
of  the  w's  are  integers,  or  may  be  placed  equal  to  the  nearest 
integer  without  involving  an  error  of  more  than  one  or  two  units 
in  the  second  significant  figure  of  any  of  the  E's.  In  the  following 
pages  E8  will  be  called  the  probable  error  of  a  standard  observa- 
tion. Obviously,  the  weight  of  a  standard  observation  is  unity 
on  the  arbitrary  scale  adopted  in  determining,  the  w's;  for,  by 
equations  (36), 


ART.  45]  MEASUREMENTS  OF  UNEQUAL  PRECISION     63 

Such  an  observation  is  not  assumed  to  have  occurred  in  any  of 
the  series  on  which  the  x's  depend,  but  is  arbitrarily  chosen  as  a 
basis  for  the  computation  of  the  weights  of  the  given  results. 

By  comparing  equations  (35)  and  (36),  we  see  that  E\  is  equal 
to  the  probable  error  of  the  arithmetical  mean  of  w\  standard 
observations.  But  it  is  also  the  probable  error  of  the  given 
result  XL  Consequently  x\  is  equivalent  to  the  arithmetical 
mean  of  wi  standard  observations.  Similar  reasoning  can  be 
applied  to  the  other  E's  and  in  general  we  have 

Xi  =  mean  of  w\  standard  observations, 
x2  =  mean  of  w2  standard  observations, 


xm  =  mean  of  wm  standard  observations. 


(i) 


The  weights  Wi,  w2}  .  .  .  wm  are  numbers  that  express  the  rela- 
tive importance  of  the  given  measurements  for  the  determination 
of  the  most  probable  value  of  the  numeric  of  the  measured  mag- 
nitude. Each  weight  represents  the  number  of  hypothetical 
standard  observations  that  must  be  combined  to  give  an  arith- 
metical mean  with  a  probable  error  equal  to  that  of  the  given 
measurement. 

45.  The  General  Mean.  —  From  equations  (i)  it  is  obvious 
that 

=  the  sum  of  Wi  standard  observations, 
=  the  sum  of  wz  standard  observations, 


wmxm  =  the  sum  of  wm  standard  observations, 

and,  consequently, 

-f  •  •  •  +  wmxm 


is  equal  to  the  sum  of  w\  +  ^2  +  .  .  •  +  WTO  standard  observa- 
tions. Since  the  probable  error  E8  is  common  to  all  of  the 
standard  observations,  they  are  equally  trustworthy  and  their 
arithmetical  mean  is  the  most  probable  value  that  we  can  assign 
to  the  numeric  X  on  the  basis  of  the  given  data.  Representing 
this  value  of  XQ  we  have 

_  WiXi  +  W2X2  +    •    •    *    +  WmXm  XQ(_V 

Wl+W2+  .  .  .  +  Wm 
The  products  W&1,  etc.,  are  called  weighted  observations  or  meas- 


64  THE   THEORY  OF  MEASUREMENTS       [ART.  45 

urements,  and  x0  is  called  the  general  or  weighted  mean.     The 
weight  WQ  of  XQ  is  obviously  given  by  the  relation 

wo  =  wi  +  w2  +  •  •  -  +  wm,  (38) 

since  XQ  is  the  mean  of  w0  standard  observations. 

Equation  (37)  for  the  general  mean  can  be  established  inde- 
pendently from  the  law  of  accidental  errors  in  the  following  manner: 
Let  coi,  o>2,  .  .  .  wm  represent  the  precision  constants  correspond- 
ing to  the  probable  errors  EI,  Ez,  •  •  •  Em,  and  let  ws  be  an 
arbitrary  quantity  connected  with  the  arbitrary  quantity  E8  by 
the  relation 

#8  =  0.2691  -• 
fc>« 

Then,  by  equations  (25)  and  (36), 

«  i2  C022  COTO2 

Wl  =  ~^>  W2  =  l^>        *-    IF-  (39) 

If  XQ  is  the  most  probable  value  of  the  numeric  X,  the  residuals 
corresponding  to  the  given  aj's  are 

ri  =  xi  —  XQ',    r2  =  xz  —  XQ',     .  .  .  rm  =  xm  —  x0. 
The  probability  that  the  true  accidental  error  of  x\  is  equal  to  r\ 


s 


in  virtue  of  equations  (39).     Similarly,  if  2/1,  2/2,  •  •  •  Vm  are  the 
probabilities  that  r\,  r2,  .  .  .  rm  are  the  true  accidental  errors  of 


•  xm} 

OJ.2 
— T-TT 

2/2  =  co2e 


Hence,  if  P  is  the  probability  that  all  of  the  r's  are  simultaneously 
equal  to  true  accidental  errors,  we  have 

w  z 

-Tr-£- 

P  ••=  (wi«o>2  .  .  .  ov)e 

and  the  most  probable  value  of  X  is  that  which  renders  P  a 
maximum.     Obviously,  the  maximum  value  of  P  occurs  when 


ART.  45]   MEASUREMENTS  OF  UNEQUAL  PRECISION     65 

(wirf  +  w2r22  +  .  .  .   +  wmrm2)  is  a  minimum.     Consequently  the 
most  probable  value  XQ  is  given  by  the  relation 

^T  (wiri2  +  w2r22  +  •  •  •  +  wmrm2)  =  0. 
Substituting  the  values  of  the  r's  and  differentiating  this  becomes 

Wi  (Xi  —  XQ)  +  W2  (X2  —  XQ)  +    •    •    •    Wm  (xm  —  XQ)   =  0. 

Hence, 

WiXi  +  W2X2  +    •    •    •    +  WmXm 

XQ  —  ; : :—          —  j 


as  given  above. 

If  we  multiply  or  divide  the  numerator  and  denominator  of 
equation  (37)  by  any  integral  or  fractional  constant,  the  value 
of  #o  is  unaltered.  Hence,  from  (36),  it  is  obvious  that  we  are  at 
liberty  to  choose  any  convenient  value  for  Ea)  whether  or  not  it 
gives  integral  values  to  the  w's.  Equations  (36)  also  show  that 
the  weights  of  measurements  are  inversely  proportional  to  the 
squares  of  their  probable  errors  and  consequently  we  may  take 

#!2  E?  EJ 

w2  =  wi-^-',    w3  =  w1^-;     .  .  .  wm  =  wi-^--         (40) 

Etf  1»»  &m 

Hence,  if  we  choose,  we  can  assign  any  arbitrary  weight  to  one  of 
the  given  measurements  and  compute  the  weights  of  the  others 
by  equation  (40). 

The  foregoing  methods  for  computing  the  weights  w\,  w2,  etc., 
are  applicable  only  when  the  given  measurements  x\,  x2,  etc.,  are 
entirely  free  from  constant  errors  and  mistakes.  When  this 
condition  is  not  fulfilled  the  method  breaks  down  because  the 
errors  of  the  x's  do  not  follow  the  law  of  accidental  errors.  In 
such  cases  it  is  sometimes  possible  to  assign  weights  to  the  given 
measurements  by  combining  the  given  probable  errors  with  an 
estimate  of  the  probable  value  of  the  constant  errors,  based  on  a 
thorough  study  of  the  methods  by  which  the  x's  were  obtained. 
Such  a  procedure  is  always  more  or  less  arbitrary,  and  requires 
great  care  and  experience,  but  when  properly  applied  it  leads  to  a 
closer  approximation  to  the  true  numeric  of  the  measured  magni- 
tude than  would  be  obtained  by  taking  the  simple  arithmetical 
mean  of  the  x's.  Since  it  involves  a  knowledge  of  the  laws  of 
propagation  of  errors  and  of  the  methods  for  estimating  the  pre- 


66  THE  THEORY  OF  MEASUREMENTS       [ART.  46 

cision  attained  in  removing  constant  errors  and  mistakes,  it  can- 
not be  fully  developed  until  we  take  up  the  study  of  the  under- 
lying principles. 

46.  Probable  Error  of  the  General  Mean.  —  When  the  given 
x's  are  free  from  constant  errors  and  the  E's  are  known,  the  weights 
of  the  individual  measurements  are  given  by  (36),  and  the  weight 
W0  of  the  general  mean  is  given  by  (38).  Consequently,  if  E0  is 
the  probable  error  of  the  general  mean,  we  have  by  analogy  with 
equations  (36) 

1*0=14     and    #0=--  (41) 


If  we  choose,  E0  may  be  expressed  in  terms  of  any  one  of  the  E's 
in  place  of  E8.  Thus,  let  En  and  wn  be  the  probable  error  and 
the  weight  of  any  one  of  the  x's,  then  by  (36) 

E> 


W 

and  eliminating  Ea  between  this  equation  and  (41)  we  have 

(42) 

When  the  weights  are  assigned  by  the  method  outlined  in  the 
last  paragraph  of  the  preceding  article,  or  when,  for  any  reason, 
the  w's  are  given  but  not  the  E's,  (41)  and  (42)  cannot  be  applied 
until  Ea  or  En  has  been  derived  from  the  given  x's  and  w's.  If 
the  number  of  given  measurements  is  large,  the  value  of  E8  corre- 
sponding to  the  given  weights  can  be  computed  with  sufficient 
precision  by  the  application  of  the  law  of  errors  as  outlined  below. 
If  the  number  of  given  measurements  is  small,  or  if  constant 
errors  and  mistakes  have  not  been  considered  in  assigning  the 
weights,  the  following  method  gives  only  a  rough  approximation 
to  the  true  value  of  Es,  and  consequently  of  EQ)  since  the  condi- 
tions underlying  the  law  of  errors  are  not  strictly  fulfilled.  It  will 
be  readily  seen  that  while  E8  may  be  arbitrarily  assigned  for  the 
purpose  of  computing  the  weights,  when  the  E's  are  given,  its 
value  is  fixed  when  the  weights  are  given. 

Let  xi,  z2,  .  .  .  xm  represent  the  given  measurements  and 
Wi,  ^2,  ...  wm,  the  corresponding  weights.  Then,  if  o?8  repre- 


ART.  46]    MEASUREMENTS  OF  UNEQUAL  PRECISION    67 

sents  the  precision  constant  of  a  standard  observation,  and  wi 
that  of  an  observation  of  weight  w\,  we  have  by  (39) 


Consequently,  if  2/A  is  the  probability  that  the  error  of  x  i  is  equal 
to  A, 


and,  by  equation  (11),  article  thirty-three,  the  probability  that 
the  error  of  x\  lies  between  the  limits  A  and  A  +  dA  is 


Now,  WiA2  is  the  weigh  ted  square  of  the  error  A,  and  in  the  follow- 
ing pages  the  product  VwA  will  be  called  a  weighted  error.  Hence, 
if  we  put  d  =  VwjA,  and  dd  =  Vw{  dA,  we  have  for  the  probability 
that  the  weighted  error  of  Xi  lies  between  the  limits  5  and  d  -\-  dd 


Since  the  same  result  would  have  been  obtained  if  we  had  started 
with  any  other  one  of  the  x's  and  w's,  it  is  obvious  that  this  equa- 
tion expresses  the  probability  that  any  one  of  the  x's,  chosen  at 
random,  is  affected  by  a  weighted  error  lying  between  the  limits 
5  and  d  +  dd.  But,  if  rid  is  the  number  of  #'s  affected  by  weighted 
errors  lying  between  these  limits,  and  m  is  the  total  number  of 
as's,  we  have  also 


or 


Hence,  the  sum  of  the  squares  of  the  weighted  errors  lying  between 
5  and  5  -f-  dd  is  given  by  the  relation 

S2us    -TO-,*    , 
=  m82-re        »dS, 


= 
"«          m 


68  THE   THEORY  OF  MEASUREMENTS       [ART.  46 

and,  by  the  method  adopted  in  articles  thirty-six  and  thirty-seven, 
we  have 


[g]  =  2  a),  r 

m         A:  Jo 


where  [52]  is  supposed  to  include  all  possible  weighted  errors 
between  the  limits  plus  and  minus  infinity.  Introducing  the 
values  of  the  S's  in  terms  of  the  w's  and  A's  this  becomes 


m  m 


which  is  an  exact  equation  only  when  the  number  of  measure- 
ments considered  is  practically  infinite. 

If  M8  is  the  mean  error  of  a  standard  observation,  we  have  from 
equation  (23) 


Hence,  from  equation  (26) 

£.  =  0.6745 


Now,  we  do  not  know  the  true  value  of  the  A's  and  the  number  of 
given  measurements  is  seldom  sufficiently  large  to  fulfill  the  con- 
ditions underlying  this  equation.  But  we  can  compute  the  gen- 
eral mean  XQ  and  the  residuals 

Ti  =  Xi  —  XQ]       r2   =  X2  —  XQ]       .    .    .    Tm  =  Xm  —  X0, 

and,  by  a  method  exactly  analogous  to  that  of  article  forty-one, 
it  can  be  shown  that  the  best  approximation  that  we  can  make  is 
given  by  the  relation 

[wr2] 


m         m  —  1 
Hence,  as  a  practicable  formula  for  computing  E8,  we  have 


Ea  =  0.6745  V-T'  (43) 

~  m  —  1 


and  consequently  E0  is  given  by  the  relation 


Eo  =  0.6745V...  r     ,,' 
in  virtue  of  equation  (41). 


ART.  47]   MEASUREMENTS  OF  UNEQUAL  PRECISION     69 


When  the  probable  errors  of  the  given  measurements  are 
known,  and  the  weights  are  computed  by  equation  (36),  the  value 
of  E8  computed  by  equation  (43)  will  agree  with  the  value  arbi- 
trarily assigned,  for  the  purpose  of  determining  the  w's,  provided 
the  x's  are  sufficiently  numerous  and  free  from  constant  errors 
and  mistakes.  The  number  of  measurements  considered  is 
seldom  sufficient  to  give  exact  agreement,  but  a  large  difference 
between  the  assigned  and  computed  values  of  E8  is  strong  evidence 
that  constant  errors  have  not  been  removed  with  sufficient  pre- 
cision. On  the  other  hand,  satisfactory  agreement  may  occur 
when  all  of  the  x's  are  affected  by  the  same  constant  error.  Con- 
sequently such  agreement  is  not  a  criterion  for  the  absence  of 
constant  errors,  but  only  for  their  equality  in  the  different  meas- 
urements. 

47.  Numerical  Example.  —  As  an  illustration  of  the  applica- 
tion of  the  foregoing  principles,  consider  the  micrometer  measure- 
ments given  under  x  in  the  following  table.  They  represent  the 
results  of  six  series  of  measurements  similar  to  that  discussed  in 
article  forty-two,  the  last  one  being  taken  directly  from  that 
article.  The  probable  errors,  computed  as  in  article  forty-two, 
are  given  under  E.  They  differ  partly  on  account  of  differences 
in  the  number  of  observations  in  the  several  series,  and  partly 
from  the  fact  that  the  individual  observations  were  not  of  the 
same  precision  in  all  of  the  series.  The  squares  of  the  probable 
errors  multiplied  by  104  are  given  under  E2  X  104  to  the  nearest 
digit  in  the  last  place  retained.  It  would  be  useless  to  carry  them 
out  further  as  the  weights  are  to  be  computed  to  only  two  signifi- 
cant figures. 


X 

E 

E*  X  10* 

w 

^5? 

w 

194.03 

0.066 

44 

11 

0.066 

193.79 

0.12 

144 

3 

0.127 

194.15 

0.091 

83 

6 

0.090 

193.85 

0.11 

121 

4 

0.110 

194.22 

0.099 

98 

5 

0.098 

194.17 

0.047 

22 

22 

0.047 

Taking  Ea  equal  to  0.22  gives  E8*  X  104  equal  to  484,  and  by 
applying  equation  (36),  we  obtain  the  weights  given  under  w  to 
the  nearest  integer.  Inverting  the  process  and  computing  the 


70 


THE   THEORY  OF  MEASUREMENTS       [ART.  47 


E's  from  the  assigned  w's  and  E8  gives  the  numbers  in  the  last 
column  of  the  table.  Since  these  numbers  agree  with  the  given 
E's  within  less  than  two  units  in  the  second  significant  figure,  we 
may  assume  that  the  approximation  adopted  in  computing  the 
w's  is  justified.  If  the  agreement  was  less  exact  and  any  of  the 
differences  exceeded  two  units  in  the  second  significant  figure,  it 
would  be  necessary  to  compute  the  w's  further,  or,  better,  to  adopt 
a  different  value  for  E8,  such  that  the  agreement  would  be  suffi- 
cient with  integral  values  of  the  w's. 

For  the  purpose  of  computation,  equation  (37)  may  be  written 
in  the  form 


XQ  =  C  + 


-  C)  +  w,  (x2  -  C)  + 


Wm  (Xm  —  C) 


where  C  is  any  convenient  number.  In  the  present  case  193  is 
chosen,  and  the  products  w  (x  —  193)  are  given  in  the  first  column 
of  the  following  table. 


w  (x  -  193) 

T 

r2  X  10< 

wr*  X  10< 

11.33 

-0.065 

42 

462 

2.37 

-0.305 

930 

2790 

6.90 

+0.055 

30 

180 

3.40 

-0.245 

600 

2400 

6.10 

+0.125 

156 

780 

25.74 

+0.075 

56 

1232 

55.84 

7844 

Substitution  in  the  above  equation  for  the  general  mean  gives 


and  this  is  the  most  probable  value  that  we  can  assign  to  the 
numeric  of  the  measured  magnitude  on  the  basis  of  the  given 
measurements. 

By  equation  (38)  the  weight,  w0,  of  the  general  mean  is  51. 
Hence  equation  (41)  gives 

0.22 


—/=• 

V51 


±0.031 


for  the  probable  error  of  x0.     Selecting  the  first  measurement 


ART.  47]   MEASUREMENTS  OF  UNEQUAL  PRECISION    71 

since  its  weight  corresponds  exactly  to  its  probable  error,  equa- 
tion (42)  gives 


Eo  =  0.066  i/      =  ±  0.031. 
»  51 

If  the  second,  third,  or  fifth  measurement  had  been  chosen,  the 
results  derived  by  the  two  formulae  would  not  have  been  exactly 
alike;  but  the  differences  would  amount  to  only  a  few  units  in  the 
second  significant  figure,  and  consequently  would  be  of  no  prac- 
tical importance.  However,  it  is  better  to  proceed  as  above  and 
select  a  measurement  whose  weight  corresponds  exactly  with  its 
probable  error  as  shown  by  the  fifth  column  of  the  first  table 
above. 

The  residuals,  computed  by  subtracting  x0  from  each  of  the 
given  measurements,  are  given  under  r  in  the  second  table;  and 
their  squares  multiplied  by  104  are  given,  to  the  nearest  digit  in 
the  last  place  retained,  under  r2  X  104.  The  last  column  of  the 
table  gives  the  weighted  squares  of  the  residuals  multiplied  by 
104.  The  sum,  [wr2],  is  equal  to  0.784.  Hence  by  equation  (43) 

E8  =  0.6745  1/0'784  =  =t  0.27, 
»      o 

and  by  equation  (44) 

JB,  =  0.6745  J^-  =  ±  0.037. 
»  51  X  o 

These  results  agree  with  the  assumed  value  of  E8  and  the  pre- 
viously computed  value  of  E0  as  well  as  could  be  expected  when 
so  small  a  number  of  measurements  are  considered.  Conse- 
quently we  are  justified  in  assuming  that  the  given  measurements 
are  either  free  from  constant  errors  or  all  affected  by  the  same 
constant  error. 

In  practice  the  second  method  of  computing  EQ  is  seldom  used 
when  the  probable  errors  of  the  given  measurements  are  known, 
since  its  value  as  an  indication  of  the  absence  of  constant  errors 
is  not  sufficient  to  warrant  the  labor  involved.  When  the  prob- 
able errors  of  the  given  measurements  are  not  known  it  is  the 
only  available  method  for  computing  EQ  and  it  is  carried  out  here 
for  the  sake  of  illustration. 


CHAPTER  VII. 
THE   METHOD   OF   LEAST   SQUARES. 

48.  Fundamental  Principles.  —  Let  Xi,  X2,  .  .  .  Xg,  and  FI, 
Y2,  .  .  .  Yn  represent  the  true  numerics  of  a  number  of  quan- 
tities expressed  in  terms  of  a  chosen  system  of  units.  Suppose 
that  the  quantities  represented  by  the  Y's  have  been  directly 
measured  and  that  we  wish  to  determine  the  remaining  quantities 
indirectly  with  the  aid  of  the  given  relations 


YZ   =   FZ  (Xl,  Xz,     .     .     .    Xq), 

Yn  =  Fn  (Xi,Xz,  •  .  .  Xq). 


(45) 


The  functions  FI,  F2,  .  .  .  Fn  may  be  alike  or  different  in  form 
and  any  one  of  them  may  or  may  not  contain  all  of  the  X's,  but 
the  exact  form  of  each  of  them  is  supposed  to  be  known. 

If  the  F's  were  known  and  the  number  of  equations  were  equal 
to  the  number  of  unknowns,  the  X's  could  be  derived  at  once 
by  ordinary  algebraic  methods.  The  first  condition  is  never  ful- 
filled since  direct  measurements  never  give  the  true  value  of  the 
numeric  of  the  measured  quantity.  Let  si;  s2,  .  .  .  sn  represent 
the  most  probable  values  that  can  be  assigned  to  the  F's  on  the 
basis  of  the  given  measurements.  If  these  values  are  substituted 
for  the  F's  in  (45),  the  equations  will  not  be  exactly  fulfilled  and 
consequently  the  true  value  of  the  X's  cannot  be  determined.  The 
differences 

Fi(Xi,XZ)  .  .  .  Xq)-si  =  k 

Fz(Xi,Xz,  .  .  .  Xq)-s2  =  k 


*,  .  .  .  Xq)-sn  =  An 


(46) 


represent  the  true  accidental  errors  of  the  s's. 

Let  Xi,  Xz,  .  .  .  xq  represent  the  most  probable  values  that  we 
can  assign  to  the  X's  on  the  basis  of  the  given  data.     Then,  since 

72 


ART.  48]      THE  METHOD  OF  LEAST  SQUARES  73 

the  s's  bear  a  similar  relation  to  the  Y's}  equations  (45)  may  be 
written  in  the  form 


Fi  (Xi,  X2)    .    .    .    Xq)   =  Sb 

F2  (xi,  x2)  .  .  .  xq)  =  s2} 
Fn  (xi,  x2}  .  .  .  xq)  =  sn, 


(47) 


where  the  functions  Fi}  F2,  etc.,  have  exactly  the  same  form  as 
before.  When  the  number  of  s's  is  equal  to  the  number  of  x's, 
these  equations  give  an  immediate  solution  of  our  problem  by 
ordinary  algebraic  methods;  but  in  such  cases  we  have  no  data 
for  determining  the  precision  with  which  the  computed  results 
represent  the  true  numerics  Xi,  X2)  etc. 

Generally  the  number  of  s's  is  far  in  excess  of  the  number  of 
unknowns  and  no  system  of  values  can  be  assigned  to  the  x's 
that  will  exactly  satisfy  all  of  the  equations  (47).  If  any  assumed 
values  of  the  x's  are  substituted  in  (47),  the  differences 

^1  (Xi,  X2)    .    .    .    Xq)  —  Si  =  7*1, 

F2  (xi,  x2)  .  .  .  xq)  -  s2  =  r2, 

Fn  (Xi,  X2,    .    .    .    Xq)   -  S-n  =  Tn 

represent  the  residuals  corresponding  to  the  given  s's.  ^Obviously,    f 
the  most  probable  values  that  we  can  assign  to  the  x's  will  be 
those  that  give  a  maximum  probability  that  these  residuals  are 
equal  to  the  true  accidental  errors  AI,  A2,  etc. 

If  the  s's  are  all  of  the  same  weight,  the  A's  all  correspond  to 
the  same  precision  constant  co.  Consequently,  as  in  article  thirty- 
five,  the  probability  that  the  A's  are  equal  to  the  r's  is 


and  this  is  a  maximum  when 

ri2  +  r22  +  .  .  .  +  rn2  =  [r2]  =  a  minimum.  (49) 

Hence,  as  in  direct  measurements,  the  most  probable  values  that 
we  can  assign  to  the  desired  numerics  are  those  that  render  the 
sum  of  the  squares  of  the  residuals  a  minimum.  For  this  reason 
the  process  of  solution  is  called  the  method  of  least  squares. 


74  THE  THEORY  OF  MEASUREMENTS       [ART.  49 

Since  the  r's  are  functions  of  the  q  unknown  quantities  xi}  x2) 
etc.,  the  conditions  for  a  minimum  in  (49)  are 


provided  the  x's  are  entirely  independent  in  the  mathematical 
sense,  i.e.,  they  are  not  required  to  fulfill  any  rigorous  mathe- 
matical relation  such  as  that  which  connects  the  three  angles  of 
a  triangle.  The  equations  (47)  are  not  such  conditions  since  the 
functions  Fi}  F2,  etc.,  represent  measured  magnitudes  and  may 
take  any  value  depending  on  the  particular  values  of  the  x's  that 
obtain  at  the  time  of  the  measurements.  When  the  r's  are  re- 
placed by  the  equivalent  expressions  in  terms  of  the  x's  and  s's  as 
given  in  (48),  the  conditions  (50)  give  q,  and  only  g,  equations 
from  which  the  x's  may  be  uniquely  determined. 

If  the  weights  of  the  s's  are  different,  the  A's  correspond  to 
different  precision  constants  coi,  0)2,  .  .  .  ,  con  given  by  the  rela- 
tions 


where  wa  is  the  precision  constant  corresponding  to  a  standard 
measurement,  i.e.,  a  measurement  of  weight  unity;  and  wi,  w2, 
.  .  .  ,  wn  are  the  weights  of  the  s's.  Under  these  conditions,  as 
in  article  forty-five,  the  most  probable  values  of  the  re's  are  those 
that  render  the  sum  of  the  weighted  squares  of  the  residuals  a 
minimum.  Thus,  in  the  case  of  measurements  of  unequal  weight, 
the  condition  (49)  becomes 

wiri2  f  w22  +  •  •  •  +  MV»2  =  [wr2]  =  a  minimum,      (51) 
and  conditions  (50)  become 

AM  =  0;     ^M  =  0;      ...  AM  =  0.      (52) 

49.  Observation  Equations.  —  The  equations  (50)  or  (52)  can 
always  be  solved  when  all  of  the  functions  FI,  F2)  .  .  .  Fn  are 
linear  in  form.  Many  problems  arise  in  practice  which  do  not 
satisfy  this  condition  and  frequently  it  is  impossible  or  incon- 
venient to  solve  the  equations  in  their  original  form.  In  such 
cases,  approximate  values  are  assigned  to  the  unknown  quantities 
and  then  the  most  probable  corrections  for  the  assumed  values 
are  computed  by  the  method  of  least  squares.  Whatever  the  form 


ART.  50]      THE  METHOD   OF  LEAST  SQUARES 


75 


of  the  original  functions,  the  relations  between  the  corrections  can 
always  be  put  in  the  linear  form  by  a  method  to  be  described  in  a 
later  chapter. 

When  the  given  functions  are  linear  in  form,  or  have  been 
reduced  to  the  linear  form  by  the  device  mentioned  above,  equa- 
tions (47)  may  be  written  in  the  form 


+  to  + 
+  to  + 


+  piXq  =  si, 
=  s2, 


pnxq  = 


(53) 


where  the  a's,  6's,  etc.,  represent  numerical  constants  given  either 
by  theory  or  as  the  result  of  direct  measurements.  These  equa- 
tions are  sometimes  called  equations  of  condition;  but  in  order 
to  distinguish  them  from  the  rigorous  mathematical  conditions, 
to  be  treated  later,  it  is  better  to  follow  the  German  practice  and 
call  them  observation  equations,  "Beobachtungsgleichungen." 

By  comparing  equations  (47),  (48),  and  (53),  it  is  obvious  that 
the  expressions 


+  to  +  CiX3  + 
-f  to  +  c2x3  + 


bnx 


cnx3 


s2  =  r2, 
pnxq  -  sn  =  rn 


(54) 


give  the  resi'duals  in  terms  of  the  unknown  quantities  x\,  xz,  etc., 
and  the  measured  quantities  si,  s2,  etc. 

50.  Normal  Equations.  —  In  the  case  of  measurements  of 
equal  weight,  we  have  seen  that  the  most  probable  values  of  the 
unknowns  x\,  x2,  etc.,  are  given  by  the  solution  of  equations  (50) 
provided  the  x's  are  independent.  Assuming  the  latter  condition 
and  performing  the  differentiations  we  obtain  the  equations 


dr,          dr. 


dr3 


dxt 


(0 


76 


THE  THEORY  OF  MEASUREMENTS       [ART.  50 


Differentiating  equations  (54)  with  respect  to  the  x's  gives 

dri  _  dr2  _ 

~dx\  ~  ai'      dxi  ~~ 


dxc 


=  an, 
=  bn, 


dr2 


and  hence  equations  (i)  become 
r2a2  +  • 
i  +  r262  +  . 


.     drn 
'•     dxq 

+  rnan  =  0, 
+  rnbn  =  0, 


(ii) 


(iii) 


-  .  .  .  +  rnpn  =  0. 

Introducing  the  expressions  for  the  r's  in  terms  of  the  x's  from 
equations  (54)  and  putting 

[aa]  =  didi  -{-  a2a2  -|-  a3a3  ~h  •  •  •  ~h  dndn} 

w> 


[as]  =  diSi  +  a2s2  +  a3s3  + 

[bd]  =  bidi  +  62a2  +  bsds  + 
[66]  =  &!&!  +  6262  +  6363  + 

[be]  =  6iCi  +  62c2  +  63c3  + 


ansn, 

6nan  =  [ab]j 

bnbn, 

6ncn 


(55) 


equations  (iii)  reduce  to 

[aa]  x-i  +  [ab]  xz  +  [ac]  x3 


[ac] 


[be]  x2  +  [cc]  x3 


[bp]xq=[bs], 

[CP]  X*   =    N, 


(56) 


giving  us  q,  so-called,  normal  equations  from  which  to  determine 
the  q  unknown  x's. 

Since  the  normal  equations  are  linear  in  form  and  contain  only 
numerical  coefficients  and  absolute  terms,  they  can  always  be 
solved,  by  any  convenient  algebraic  method,  provided  they  are 
entirely  independent,  i.e.,  provided  no  one  of  them  can  be  ob- 
tained by  multiplying  any  other  one  by  a  constant  numerical 


ART.  50]      THE   METHOD   OF   LEAST   SQUARES  77 

factor.  This  condition,  when  strictly  applied,  is  seldom  violated 
in  practice;  but  it  occasionally  happens  that  one  of  the  equations 
is  so  nearly  a  multiple  or  submultiple  of  another  that  an  exact 
solution  becomes  difficult  if  not  impossible.  In  such  cases  the 
number  of  observation  equations  may  be  increased  by  making 
additional  measurements  on  quantities  that  can  be  represented 
by  known  functions  of  the  desired  unknowns.  The  conditions 
under  which  these  measurements  are  made  can  generally  be  so 
chosen  that  the  new  set  of  normal  equations,  derived  from  all  of 
the  observation  equations  now  available,  will  be  so  distinctly 
independent  that  the  solution  can  be  carried  out  without  difficulty 
to  the  required  degree  of  precision. 

By  comparing  equations  (53)  and  (56),  it  is  obvious  that  the 
normal  equations  may  be  derived  in  the  following  simple  manner. 
Multiply  each  of  the  observation  equations  (53)  by  the  coefficient 
of  xi  in  that  equation  and  add  the  products.  The  result  is  the 
first  normal  equation.  In  general,  q  being  any  integer,  multiply 
each  of  the  observation  equations  by  the  coefficient  of  xq  in  that 
equation  and  add  the  products.  The  result  is  the  gth  normal 
equation.  The  form  of  equations  (56)  may  be  easily  fixed  in 
mind  by  noting  the  peculiar  symmetry  of  the  coefficients.  Those 
in  the  principal  diagonal  from  left  to  right  are  [aa],  [66],  [cc],  etc., 
and  coefficients  situated  symmetrically  above  and  below  this 
diagonal  are  equal. 

When  the  given  measurements  are  not  of  equal  weight,  the 
observation  equations  (53),  and  the  residual  equations  (54)  remain 
unaltered,  but  the  normal  equations  must  be  derived  from  (52) 
in  place  of  (50).  Since  the  weights  Wi,  w2,  etc.,  are  independent 
of  the  x's,  if  we  treat  equations  (52)  in  the  same  manner  that  we 
have  treated  (50),  we  shall  obtain  the  equations 

•  *  +  wnrnan  =  0, 
•'•.  .4  Wn&n  =  0, 


(iv) 

+  Wtfzpz  +    '    '    '     +  Wnrnpn  =  0, 

in  place  of  equations  (iii).     Hence,  if  we  put 

[iWia]  =  Wididi  -f~  WzClzCLz  ~\~    '    '    '    ~\~  WndnCLnj 

(57) 


[was]  =  WidiSi  +  w&zSz  +  •  •  •  +  wnansnj 

'    •    -\-WnpnPn, 


78 


THE  THEORY  OF  MEASUREMENTS       [ART.  51 


the  normal  equations  become 
[waa]  xi  +  [wab]  x2  +  [wac]  z3 
[wab]  Xi  +  [wbb]  xz  +  [wbc]  xz 
[wac]  X!  +  [wbc]  x2  +  [wcc]  xz 


+  [wap]  xq  =  [was], 
+  [wbp]  xq  =  [wbs], 
+  [wcp]  xq  =  [wcs], 


(58) 


[wap]xi  +  [wbp]x2  +  [wcp]x$  +  •  •  •  +  [wpp]xq  =  [wps]. 

These  equations  are  identical  in  form  with  equations  (56),  and 
they  may  be  solved  under  the  same  conditions  and  by  the  same 
methods  as  those  equations.  Consequently,  in  treating  methods 
of  solution,  we  shall  consider  the  measurements  to  be  of  equal 
weight  and  utilize  equations  (56).  All  of  these  methods  may  be 
readily  adapted  to  measurements  of  unequal  weight  by  substitut- 
ing the  coefficients  as  given  in  (57)  for  those  given  in  (55). 

51.  Solution  with  Two  Independent  Variables.  —  When  only 
two  independent  quantities  are  to  be  determined  the  observation 
equations  (53)  become 

" 


=  s, 
and  the  normal  equations  (56)  reduce  to 

[aa]  Xi  +  [ab]  x2  =  [as], 

[ab]  X!  +  [bb]  x2  =  [bs]. 
Solving  these  equations  we  obtain 

[bb]  [as]  -  [ab]  [bs] 

[aa]  [bb]  -  [ab]2 
_  [aa]  [bs]  —  [ab]  [as] 

[aa]  [bb]  -  [ab]2 

As  an  illustration,  consider  the  determination  of  the  length  Z/0 
at  0°  C.,  and  the  coefficient  of  linear  expansion  a  of  a  metallic 
bar  from  the  following  measurements  of  its  length  Lt  at  temper- 
ature t°  C. 


(56a) 


(59) 


t 

Lt 

C. 

20 

mm. 
1000.36 

30 

1000.53 

40 

1000.74 

50 

1000.91 

60 

1001.06 

Ara.51]      THE  METHOD  OF  LEAST  SQUARES 


79 


or 


Within  the  temperature  range  considered,  Lt  and  t  are  connected 
with  LO  and  a  by  the  relation 

Lt  =  Lo  (1  +  at), 

Lt  =  Lo  +  L0at,  (v) 

and  a  set  of  observation  equations  might  be  written  out  at  once 
by  substituting  the  observed  values  of  Lt  and  t  in  this  equation. 
But  the  formation  of  the  normal  equations  and  the  final  solution 
is  much  simplified  when  the  coefficients  and  absolute  terms  in  the 
observation  equations  are  small  numbers  of  nearly  the  same  order 
of  magnitude.  To  accomplish  this  simplification,  the  above  func- 
tional relation  may  be  written  in  the  equivalent  form 


and  if  we  put 


it  becomes 


Lt  -  1000  =  Lo  -  1000  +  WL<xx  — 

Lt  -  1000  =  s;     JQ  =  6, 

LO —  1000  =  Xi]     10  LOCK  =  Xz, 
Xi  -J-  6^2  =  s. 


(vi) 


Using  this  function,  all  of  the  a's  in  equation  (53a)  become  equal 
to  unity  and  the  6's  and  s's  may  be  computed  from  the  given 


observations  by  equations   (vi). 

the  observation  equations  are 

xi  +  2  z2  = 
xi  +  3  x2  = 


Hence,   in  the  present  case, 


.36, 
.53, 

xl+±x2=    .74, 

zi  +  5z2  =    .91, 

Xl  +  6x2  =  1.06. 

For  the  purpose  of  forming  the  normal  equations,  the  squares 
and  products  of  the  coefficients  and  absolute  terms  are  tabulated 
as  follows  : 


Obs. 

aa 

ab 

as 

bb 

bs 

1 

2 

0.36 

4 

0.72 

2 

3 

0.53 

9 

1.59 

3 

4 

0.74 

16 

2.96 

4 

5 

0.91 

25 

4.55 

5 

6 

1.06 

36 

6.36 

5 

20 

3.60 

90 

16.18 

[aa] 

W 

[as] 

[bb] 

[bs] 

Substituting  these  values  of  the  coefficients  in  (56a)  gives  the 
normal  equations 


80 


THE  THEORY  OF  MEASUREMENTS       [ART.  51 


=    3.60, 
=  16.18, 

and  by  (59)  we  have 

_  90  X  3.60  -  20  X  16.18 

5  X  90  -  400 
5  X  16.18  -  20  X  3.60 


=  0.008, 
=  0.178. 


5  X  90  -  400 

From  these  results,  with  the  aid  of  relations  (vi),  we  find 
Lo  =  xi  +  1000  =  1000.008, 

L0a  =  ^  =  0.0178, 


0.0178 


=  0.0000178, 


and  finally 

Lt  =  1000.008  (1  +•  0.0000178 1)  millimeters.  (vii) 

The  differences  between  the  values  of  Lt  computed  by  equation 
(vii),  and  the  observed  values  give  the  residuals.  But  they  can 
be  more  simply  determined  by  using  the  above  values  of  x\ 
and  Xz  in  the  observation  equations  and  taking  the  difference 
between  the  computed  and  observed  values  of  s.  Thus,  if  s' 
represents  the  computed  value  and  r  the  corresponding  residual 

s'  =  0.008  +  0.178  6, 
and  r  =  sf  —  s. 

With  the  values  of  s  and  6  used  hi  the  observation  equations  we 
obtain  the  residuals  as  tabulated  below: 


s' 

8 

r 

7-2  X   10* 

0.364 
0.542 
0.720 
0.898 
1.076 

0.36 
0.53 
0.74 
0.91 
1.06 

+0.004 
+0.012 
-0.020 
-0.012 
+0.016 

0.16 
1.44 
4.00 
1.44 
2.56 

[r2]  =  9.60XlO~4 

Since  the  above  values  of  x\  and  x2  were  computed  by  the  method 
of  least  squares,  the  resulting  value  of  [r2],  i.e.,  .000960,  should  be 
less  than  that  obtainable  with  any  other  values  of  x\  and  x%. 
That  this  is  actually  the  case  may  be  verified  by  carrying  out  the 
computation  with  any  other  values  of  x\  and  xz. 


ART.  52]      THE  METHOD  OF  LEAST  SQUARES 


81 


52.  Adjustment  of  the  Angles  About  a  Point.  —  As  an  illus- 
tration of  the  application  of  the  method  of  least  squares  to  the 
solution  of  a  problem  involving  more  than  two  unknown  quanti- 
ties, suppose  that  we  wish  to  determine  the  most  probable  value 
of  the  angles  AI,  AZ,  and  A 3,  Fig.  9,  from  a  series  of  independent 
measurements  of  equal  weight  on  the  angles  Mi,  M2,  .  .  .  M6. 
If  the  given  measurements  were  all  exact,  the  equations 

AI  =  Mi;    AZ  =  M2;    A3  =  M3; 

AI-\-  AZ  =  M4;    AI  +  AZ  -{-As  =  MS;     and  Az  -\- As  =  Me, 

would  all  be  fulfilled  identically.  In  practice  this  is  never  the 
case  and  it  becomes 
necessary  to  adjust  the 
values  of  the  A's  so  that 
the  sum  of  the  squares 
of  the  discrepancies  will 
be  a  minimum.  The 
adjustment  may  be  ef- 
fected by  adopting  the 
above  equations  as  ob- 
servation equations  and 
proceeding  at  once  to 
the  solution  for  the  A's 
by  the  method  of  least 
squares.  But  the  ob- 
served values  of  the  M's 
usually  involve  so  many 
significant  figures  that 
the  computation  would 
be  tedious.  It  is  better 
to  adopt  approximate 
values  for  the  A's  and  then  compute  the  necessary  corrections  by 
the  method  of  least  squares. 

For  this  purpose,  suppose  we  adopt  MI,  M2,  and  M3  as  approxi- 
mate values  of  A\,  A2,  and  As  respectively  and  let  xi,  Xz,  and  x3 
represent  the  corrections  that  must  be  applied  to  the  M's  in  order 
to  give  the  most  probable  values  of  the  A's.  Then,  putting 

AI  =  MI  +  xi,    AZ  =  MZ  +  Xz,    and  A3  =  M3  +  #3,     (viii) 
the  above  equations  become 


FIG.  9. 


82 


THE  THEORY  OF  MEASUREMENTS       [ART.  52 


+  x2 


=  0, 
=  0, 
=  0, 
=  M4  -  (Af !  +  M2), 


To  render  the  problem  definite,  suppose  that  the  following 
values  of  the  M's  have  been  determined  with  an  instrument  read- 
ing to  minutes  of  arc  by  verniers: 

Mi  =  10°  49'.5,  M4  =  45°  24'.0, 

M2  =  34°  36'.0,  M6  =  60°  53'.5, 

M3  =  15°  25'.5,  M6  =  50°    O'.O. 

Substituting  these  values  in  the  above  equations  we  obtain 
xi  =  0, 

x2          =  0, 


2'.5, 


Adopting  these  as  our  observation  equations  and  comparing  with 
(53)  we  obtain  the  coefficients  and  absolute  terms  tabulated  below: 


Oba. 

a 

b 

c 

s 

1 

1 

0 

0 

0 

2 

0 

1 

0 

0 

3 

0 

0 

1 

0 

4 

1 

1 

0 

-1.5 

5 

1 

1 

1 

2.5 

6 

0 

1 

1 

-1.5 

The  squares  and  products  of  the  coefficients  and  absolute  terms 
may  be  tabulated,  for  the  purpose  of  forming  the  normal  equations, 
as  follows : 


M 

ab 

ac 

as 

66 

be 

bs 

cc 

cs 

1 

0 
0 

1 
1 

0 

0 
0 
0 

1 
1 

0 

0 
0 
0 
0 

1 

0 

0 
0 
0 
-1.5 
2.5 
0 

0 
0 

1 
1 
1 

0 
0 
0 
0 

1 
1 

2 

[be] 

0 
0 
0 
-1.5 
2.5 
-1.5 

0 
0 

1 

0 

1 

1 

0 
0 
0 
0 
2.5 
-1.5 

3 

[aa] 

[ab] 

1 

[ac] 

1 

fas] 

4 
[66] 

-0.5 

[6s] 

3 

[cc] 

1 

[cs] 

ART.  53]      THE  METHOD   OF  LEAST  SQUARES  83 

Substituting  these  values  in  (56)  the  three  normal  equations 
become 


-0.5, 
1  xi  -f  2  x2  +  3  z3  =  1, 

and  solution  by  any  method  gives 

xi  =  0.625;    x2  =  -  0.75;    x3  =  0.625. 

With  these  results  together  with  the  given  values  of  MI,  Mz, 
and  M3  we  obtain  from  equations  (viii) 

A!  =  10°  50M25, 
A2  =  34°  35'.25, 
A3  =  15°  26M25. 

In  a  problem  so  simple  as  the  present  the  normal  equations  are 
generally  written  out  at  once  from  the  observation  equations  by 
the  rule  stated  in  article  fifty,  without  taking  the  space  and  time 
to  tabulate  the  coefficients,  etc.  But,  until  the  student  is  thor- 
oughly familiar  with  the  process,  it  is  well  to  form  the  tables  as 
a  check  on  the  computations  and  to  make  sure  that  none  of  the 
coefficients  or  absolute  terms  have  been  omitted.  For  this  reason 
the  tabulation  has  been  given  in  full  above  and  the  student  is 
advised  to  carry  out  the  formation  of  the  normal  equations  by 
the  shorter  method  as  an  exercise. 

53.  Computation  Checks.  —  When  the  number  of  unknowns 
is  greater  than  two  and  a  large  number  of  observation  equations 
are  given  with  coefficients  and  absolute  terms  involving  more  than 
two  significant  figures,  the  formation  of  the  normal  equations  is 
the  most  tedious  and  laborious  part  of  the  computations.  It  is, 
therefore,  advantageous  to  devise  a  means  of  checking  the  com- 
puted coefficients  and  absolute  terms  in  the  normal  equations 
before  we  proceed  to  the  final  solution. 

For  this  purpose  compute  the  n  quantities  t\t  ^2,  ...  tn  by  the 
equations 

ai  +  &i  -f  ci  +  •  •  -  +  pi  =  ti,~ 

02  +  &2  4-  c2  -f  •  •  •  +  pz  =  h, 


On  +  &„  +  Cn  -f    -    •    •    +  pn  = 


(60) 


84  THE   THEORY  OF  MEASUREMENTS       [ART.  54 

where  the  a's,  b's,  etc.,  are  the  coefficients  in  the  given  observa- 
tion equations.  Multiply  the  first  of  equations  (60)  by  Si,  the 
second  by  s2,  etc.,  and  add  the  products.  The  result  is 

[as]  +  [bs]  +  [cs]  +  •  •  •  +  \ps]  =  [ts].  (61) 

In  the  same  way,  multiplying  by  the  a's  in  order  and  adding,  then 
by  the  b's  in  order  and  adding,  etc.,  we  obtain  the  following  rela- 
tions 

[aa]  +  [db]  +  [ac]  +  ••.-.  +  [ap]  =  [at], 

[ab]  +  [bb]  +  [be]  +  ••••  +  [bp]  =  [bt], 

[ac]  +  [be]  +  [cc]  +  •  •  •  +  [cp]  =  [ct],  (62) 

[ap]  +  \bp]  +  [ep]  +  .  .  .  +  \pp]  =  \pt]. 

If  the  absolute  terms  in  the  normal  equations  have  been  accu- 
rately computed,  equation  (61)  reduces  to  an  identity.  If  the 
coefficients  have  been  accurately  computed  equations  (62)  all 
become  identities.  Hence  (61)  is  a  check  on  the  computation  of 
the  absolute  terms  and  equations  (62)  bear  the  same  relation  to 
the  coefficients.  The  extra  labor  involved  in  computing  the  quan- 
tities [ts]t  [at],  .  .  .  ,  [pt]  is  more  than  repaid  by  the  added  confi- 
dence in  the  accuracy  of  the  normal  equations. 

When  all  attainable  significant  figures  are  retained  throughout 
the  computations,  the  checks  (61)  and  (62)  should  be  identities. 
In  practice  the  accuracy  of  the  measurements  is  seldom  sufficient 
to  warrant  so  extensive  a  use  of  figures,  and,  consequently,  the 
squares  and  products,  aa,  ab,  .  .  .  as,  at,  etc.,  are  rounded  to  such 
an  extent  that  the  computed  values  of  the  x's  will  come  out  with 
about  the  same  number  of  significant  figures  as  the  given  data. 
Judgment  and  experience  are  necessary  in  determining  the  number 
of  significant  figures  that  should  be  retained  in  any  particular 
problem  and  it  would  be  difficult  to  state  a  general  rule  that 
would  not  meet  with  many  exceptions.  When  the  computed 
coefficients  and  absolute  terms  are  rounded,  as  above,  the  checks 
may  not  come  out  absolute  identities,  but  they  should  not  be 
accepted  as  satisfactory  when  the  discrepancy  is  more  than  two 
units  in  the  last  place  retained. 

54.  Gauss's  Method  of  Solution.  —  When  the  normal  equa- 
tions (56)  are  entirely  independent,  they  may  be  solved  by  any 
of  the  well-known  methods  for  the  solution  of  simultaneous 
linear  equations  and  lead  to  unique  values  of  the  unknown  quan- 


ART.  54]      THE   METHOD   OF   LEAST  SQUARES  85 

titles  xi,  x2)  etc.  Gauss's  method  of  substitution  is  frequently 
adopted  for  this  purpose  since  it  permits  the  computation  to  be 
carried  out  in  symmetrical  form  and  provides  numerous  checks 
on  the  accuracy  of  the  numerical  work.  The  general  principles 
of  the  method  will  be  illustrated  and  explained  by  completely 
working  out  a  case  in  which  there  are  only  three  unknowns. 
Since  the  process  of  solution  is  entirely  symmetrical,  it  can  be 
easily  extended  for  the  determination  of  a  larger  number  of 
unknowns,  but  too  much  space  would  be  required  to  carry  through 
the  more  general  case  here. 

When  only  three  unknowns  are  involved,  the  normal  equations 
(56)  and  the  check  equations  (60)  and  (61)  may  be  completely 
written  out  in  the  following  form,  the  computed  quantities  and 
equations  being  placed  at  the  left,  and  the  checks  at  the  right. 

[aa]  xi  +  [ab]  x2  +  [ac]  x3  =  [as].         [aa]  +  [ab]  +  [ac]  =  [at]. 

[ab]  xi  +  [bb]  x2  +  [be]  x3  =  [bs].         [ab]  +  [bb]  +  [be]  =  [bt]. 

[ac]  xi  +  [be]  x2  +  [cc]  x3  =  [cs].         [ac]  +  [be]  +  [cc]  =  [ct]. 

[as]  +  [bs]+[cs]  =[st].\ 

Solve  the  first  equation  on  the  left  for  xiy  giving 
[as]       [ab]          [ac] 

Xi  =  7 7  —  f 1  X2  —  f 1  X$. 

[aa]       [aa]          [aa\ 
Compute  the  following  auxiliary  quantities: 


(63) 


[56]  _  P4  [0&]  =  [bb  •  1],  [bt]  -  pi  M  =  [^ '  1L 

L  aa] 

[a61 


[6c]- 


M  L"  M 


~       M  =  [6s  '  1]'         M  ~      N1  =  [st 


As  a  check  on  these  computations  we  notice  that 

[bb  •  1]  +  [be  -  1]  =  [bb]  +  [be]  -  |^|  ([ab]  +  [«c]), 

[aaj 

=  [bt]  -  lab]  -         ([at]  -  [aa]), 


86  THE   THEORY  OF  MEASUREMENTS       [ART.  54 

In  a  similar  way  we  may  show  that  we  should  have 

[6c-l]  +  [cc-l]  =  [cM]    and     [6s-  1]  +  [cs-  1]  =  [st-  1]. 

Substituting  (64)  in  the  last  two  of  (63)  and  placing  the  above 
checks  to  the  right,  we  have  the  equations 
[bb  -I]x2  +  [be  •  1]  xs  =  [bs  •  1],       [bb  •  1]  +  [be-  1]  =  [fa  •  1], 
[be  -I]x2  +  [cc  •  1]  z3  =  [cs  •  1],        [be  •  1]  +  [cc-  1]  =  [ct  •  1],      (65) 

[6s-l]  +  [cs.l]  =  [s*.  1],. 

which  show  the  same  type  of  symmetry  as  (63),  but  contain  only 
two  unknown  quantities.     Solve  the  first  of  (65)  for  x2  giving 


__ 
*2~[6&.l]      [bb-lf3' 

and  compute  the  following  auxiliaries: 

[<*•!]  -  [l^jlfc-1]  =  [«-2],        [cM]  -  l~^}[bt.  1}  =  let-  2], 

(cs  •  1}  -  |^jj  [bs  •  1}  -  [cs  •  2],        [st  •  1]  -  |^|j  (bt  •  1]  =  [*  •  2}. 

By  a  method  similar  to  that  used  above  we  can  show  that  we 
should  have 

[cc  •  2]  =  [ct  •  2]     and     [cs  •  2]  =  [st  •  2]. 

Hence,  substituting  (66)  in  the  last  of  (65),  we  have 
[cc  •  2]  x3  =  [cs  •  2],       [cc  •  2]  =  [ct  •  2], 
[cs.2]  =  N-2], 
and  consequently 

[cs  •  2]  _. 

*"fc^t'  (67) 

Having  determined  the  value  of  x3  from  (67),  x%  may  be  cal- 
culated from  (66),  and  then  Xi  from  (64). 

A  very  rigorous  check  on  the  entire  computation  is  obtained  as 
follows:  using  the  computed  values  of  Xi,  xz,  and  z3  in  equations 
(54),  derive  the  residuals 


(68) 


-  s2, 

Tn  =  dnXi  ~|-  OnX2  ~\-  CnXs         Sn, 

and  then  form  the  sums 

[rr]  =  n2  +  r22  +  r32  +  -  -  -  +  rn2, 

[SS]   =  Si2  +  S22  +  S32  +    •    •    •    +  «n2. 


ART.  55]      THE   METHOD   OF   LEAST   SQUARES  87 

If  the  computations  are  all  correct,  the  computed  quantities  will 
satisfy  the  relation 

W  =  M-[aS]-M[6S.l]-[cs.2].       (69) 


To  prove  this,  multiply  the  first  of  (68)  by  ri,  the  second  by  r%, 
etc.,  and  add  the  products.     The  result  is 

[rr]  =  [ar]  Xi  +  [br]  x2  -f  [cr]  £3  -  [sr]. 
But  from  equations  (iii),  article  fifty, 

[ar]  =  [br]  =  [cr]  =  0, 
consequently 

[rr]=-  [«•].  (70) 

Multiply  each  of  equations  (68)  by  its  s;   add,  taking  account  of 
(70),  and  we  obtain 

[rr]  =  [ss]  -  [as]  Xi  -  [6s]  xz  -  [cs]  xz. 

Eliminating  x\,  X2,  and  z3,  in  succession  with  the  aid  of  (64),  (66), 
and  (67)  we  find 

[rr]  =  [ss]  -         [as]  -  [6s  •  1]  x2  -  [cs  •  1]  x9, 


and  finally 

r     i         r     i         las]  r     i         [&s  *  1]  n       n         tcs  '  2]  r         Ol 

[rr]  =  M  ~  y  M  -  I667i]  [6s  '  1]  -  RT2]  [cs  '  2]' 

which  is  identical  with  (69). 

55.  Numerical  Illustration  of  Gauss's  Method.  —  The  fore- 
going methods  are  most  frequently  used  for  the  adjustment  of 
astronomical  and  geodetic  observations,  and  their  application  to 
particular  problems  is  fully  discussed  in  practical  treatises  on 
such  observations.  The  physical  problems,  to  which  they  are 
applicable,  usually  involve  the  determination  of  an  empirical 
relation  between  mutually  varying  quantities.  Such  problems 
will  be  discussed  at  some  length  in  Chapter  XIII,  and  the  corre- 
sponding observation  equations  will  be  developed. 

It  would  require  too  much  space  to  carry  out  the  complete  dis- 
cussion of  such  a  problem,  in  this  place,  with  all  of  the  observa- 
tions made  in  any  actual  investigation.  But,  for  the  purpose  of 
illustration,  the  most  probable  values  of  xi,  Xz,  and  x3  will  be 


88 


THE   THEORY  OF  MEASUREMENTS       [ART.  55 


derived,  from  the  following  typical  observation  equations,   by 
Gauss's  method  of  solution: 


+  2x2+  0.4z3  = 
+  4x2  +  1.6x3  = 
+  6  x2  +  3.6  z8  = 
+  8x2  +  6.4x3  = 
+10x2  +10.0^3  = 


0.24, 

-  1.18, 

-  1.53, 

-  0.69, 
1.20, 
4.27. 


Since  the  coefficient  of  xi  is  unity  in  each  of  these  equations, 
the  products  aa,  ab,  aCj  as,  and  at  are  equal  to  a,  6,  c,  s,  and  t, 
respectively.  Consequently  the  first  five  columns  of  the  follow- 
ing table  show  the  coefficients,  absolute  terms,  and  check  terms 
(t  =  a  +  b  +  c)  of  the  observation  equations  as  well  as  the 
squares  and  products  indicated  at  the  head  of  the  columns.  The 
sums  [aa],  [ab],  etc.,  are  given  at  the  foot  of  the  columns  and  the 
checks,  by  equations  (61)  and  (62),  are  given  below  the  tables. 
In  the  present  case,  the  coefficients  are  expressed  by  so  few  signifi- 
cant figures  that  it  is  not  necessary  to  round  the  computed  products 
and  consequently  the  checks  come  out  identities. 


aa 

ab 

ac 

as 

at 

bb 

be 

0 
2 
4 
6 
8 
10 

0.0 
0.4 
1.6 
3.6 
6.4 
10.0 

0.24 
-1.18 
-1.53 
-0.69 
1.20 
4.27 

1.0 
3.4 
6.6 
10.6 
15.4 
21.0 

0 
4 
16 
36 
64 
100 

0.0 
0.8 
6.4 
21.6 
51.2 
100.0 

6 

M 

30 
[ab] 

22.0 

M 

2.31 
[as] 

58.0 

M 

220 

m 

180.0 
[be] 

Check:  [ 

aa]  +  [ab]  +  [c 

ic]  =  58.0. 

bs 

cc 

cs 

bl 

ct 

st 

0.00 
-2.36 
-6.12 
-4.14 
9.60 
42.70 

0.00 
0.16 
2.56 
12.96 
40.96 
100.00 

0.00 
-0.472 
-2.448 
-2.484 
7.680 
42.700 

0.0 
6.8 
26.4 
63.6 
123.2 
210.0 

0.00 
1.36 
10.56 
38.16 
98.56 
210.00 

0.24 
-  4.012 
-10.098 
-  7.314 

18.480 
89.670 

39.68 
[bs] 

156.64 
[cc] 

44.976 
[cs] 

430.0 

M 

358.64 
[ct] 

86.966 
[st] 

Checks:  [ab]  +  [66]  +  [be]  =  430.0 
[ac]  +  [be]  +  [cc]  =         358.64 
M  +  [6s]  +  [cs]=          86.966 

ART.  55]      THE   METHOD   OF   LEAST   SQUARES 


89 


The  normal  equations  and  their  checks  might  now  be  written 
out  in  the  form  of  equations  (63),  but,  since  the  coefficients  and 
other  data  necessary  for  their  solution  are  all  tabulated  above,  it 
is  scarcely  worth  while  to  repeat  the  same  data  in  the  form  of 
equations.  The  computation  of  the  auxiliaries  [bb  •  1],  [be  •  1], 
etc.,  and  the  final  solution  for  xi}  x2)  and  #3  by  logarithms  is  best 
carried  out  in  tabular  form  as  illustrated  on  pages  90  and  91. 
The  meaning  of  the  various  quantities  appearing  in  these  tables,  and 
the  methods  by  which  they  are  computed,  will  be  readily  under- 
stood by  comparing  the  numerical  process  with  the  literal  equa- 
tions of  the  preceding  article.  When  the  letter  n  appears  after  a 
logarithm  it  indicates  that  the  corresponding  number  is  to  be  taken 
negative  in  all  computations. 

The  computation  of  the  residuals  by  equations  (68)  and  the 
final  check  by  (69)  is  carried  out  in  the  following  table,  where 
Scale,  is  written  for  the  value  of  the  expression  axi  +  bx2  +  cxs, 
when  the  computed  values  of  x\,  x2,  and  x3  are  used  and  s0bs.  is 
the  corresponding  value  of  s  in  the  observation  equations.  Thus 


+- 


—  Si  =  Si  calc.  ~  Si  obs.- 


I* 

SObB. 

r 

*Xio. 

ss 

0.245 
-1.195 
-1.512 
-0.709 
1.215 
4.264 

0.24 
-1.18 
-1.53 
-0.69 
1.20 
4.27 

+0.005 
-0.015 
+0.018 
-0.019 
+0.015 
-0.006 

25 
225 
324 
361 
225 
36 

0.0576 
1.3924 
2.3409 
0.4761 
1.4400 
18.2329 

.001196 
[rr] 

23.9399 

[as]r    ,           [6s  •  1]  ,      .,,                     [cs  •  2] 

:s-2] 

52           =         23.9387 
0.0012 

[aa\                 [oo  •  i\                               [cc  •  A\ 
0.8893     +         11.3042             +               11.74 
Final  check  by  (69):                    [rr] 

Since  the  checks  are  all  satisfactory,  we  are  justified  in  assum- 
ing that  the  computations  are  correct.  Hence  the  most  probable 
values  of  the  unknowns,  derivable  from  the  given  observation 
equations,  are 

xi  =  0.245;    x2=  -  1.0003;    z3  =  1.4022, 


90 


THE   THEORY  OF  MEASUREMENTS       [ART.  55 


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ART.  55]       THE    METHOD    OF   LEAST  SQUARES  91 


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92  THE   THEORY  OF  MEASUREMENTS        [ART.  56 

and  the  corresponding  empirical  relation  becomes 
s  =  0.245  a  -  1.0003  6  +  1.4022  c. 

A  small  number  of  observation  equations  with  simple  coefficients 
have  been  chosen,  in  the  above  illustration,  partly  to  save  space 
and  partly  in  order  that  the  computations  may  be  more  readily 
followed.  In  practice  it  would  seldom  be  worth  while  to  apply 
the  method  of  least  squares  to  so  small  a  number  of  observations 
or  to  adopt  Gauss's  method  of  solution  with  logarithms  when  the 
normal  equations  are  so  simple.  When  the  number  of  observa- 
tions is  large  and  the  coefficients  involve  more  than  three  or  four 
significant  figures,  the  method  given  above  will  be  found  very 
convenient  on  account  of  the  numerous  checks  and  the  symmetry 
of  the  computations.  In  order  to  furnish  a  model  for  more 
complicated  problems,  the  process  has  been  carried  out  completely 
even  in  the  parts  where  the  results  might  have  been  foreseen 
without  the  use  of  logarithms. 

56.  Conditioned  Quantities.  —  When  the  unknown  quantities, 
Xi,  Xz,  etc.,  are  not  independent  in  the  mathematical  sense,  the 
foregoing  method  breaks  down  since  the  equations  (50)  no  longer 
express  the  condition  for  a  minimum  of  [rr].  In  such  cases  the 
number  of  unknowns  may  be  reduced  by  eliminating  as  many  of 
them  as  there  are  rigorous  mathematical  relations  to  be  fulfilled. 
The  remaining  unknowns  are  independent  and  may  be  deter- 
mined as  above.  The  eliminated  quantities  are  then  determined 
with  the  aid  of  the  given  mathematical  conditions. 

For  the  purpose  of  illustration,  consider  the  case  of  a  single 
rigorous  relation  between  the  unknowns,  and  let  the  correspond- 
ing mathematical  condition  be  represented  by  the  equation 

0(xltx,,  .  .  .  ,  xq)  =0.  (71) 

As  in  the  case  of  unconditioned  quantities,  the  observation  equa- 
tions (53)  are 

+  C&s  +    •    •    •     +  piXq  =  Si, 


cnx3       •  •  •       pnxq  = 

The  solution  of  (71)  for  x\,  in  terms  of  Xz,  xs,  .  .  .,  xq,  may  be 
written  in  the  form 

xi=f(xz,xa,  ..*,*«)•  (72) 


ART.  56]      THE   METHOD  OF  LEAST  SQUARES  93 

Introducing  this  value  of  xi,  equations  (53)  become 

+  ClX*  +    •    *    •    +  PlXq  =  Si, 
+  C2X3  +    •    •    •     +  P2Za  =  S2, 

4-  cnz3  +  •  •  •  +  pnxq  =  s». 

Since  the  form  of  6  is  known,  that  of  /  is  also  known.  Hence,  by 
collecting  the  terms  in  x%,  xS}  etc.,  and  reducing  to  linear  form, 
if  necessary,  we  have 

bixz  +  ci'xs  +  •  •  •  +  p\xq  =  s/, 


The  x's  in  these  equations  are  independent,  and,  consequently, 
they  may  be  determined  by  the  methods  of  the  preceding  articles. 
Using  the  values  thus  obtained  in  (71)  or  (72)  gives  the  remaining 
unknown  x\.  The  #'s,  thus  determined,  obviously  satisfy  the 
mathematical  condition  (71)  exactly,  -and  give  the  least  magnitude 
to  the  quantity  [rr]  that  is  consistent  with  that  condition.  They 
are,  consequently,  the  most  probable  values  that  can  be  assigned 
on  the  basis  of  the  given  data. 

As  a  very  simple  example,  consider  the  adjustment  of  the 
angles  of  a  plane  triangle.  Suppose  that  the  observed  values  of 
the  angles  are 

si  =  60°  1';     s2  =  59°  58';    s3  =  59°  59'. 

The  adjusted  values  must  satisfy  the  condition 

xi  +  x2  +  x*  =  180°, 
or 

xi  =  180°  -  x2  -  x3. 

Eliminating  Xi  from  the  observation  equations, 
xi  =  Si't    Xz  =  s2;    and     xs  —  s3; 
and  substituting  numerical  values  we  have 
xz+x3  =  119°  59', 
x2  =    59°  58', 

x3  =    59°  59'. 

The  corresponding  normal  equations  are 
2z2  +  z3  =  179°  57', 
=  179°  58', 


94  THE   THEORY  OF  MEASUREMENTS       [ART.  56 

from  which  we  find 

x2  =  59°  58'.7  and  xs  =  59°  59'.7. 

Then,  from  the  equation  of  condition, 

xi  =  60°  1'.6. 

When  there  are  two  relations  between  the  unknowns,  expressed 
by  the  equations 

01  (xi,  xt,  .  .  .  ,  xq)  =  0, 

02  (xi,  x2,  .  .  .  ,  xq)  =  0, 

they  may  be  solved  simultaneously  for  xi  and  x2,  in  terms  of  the 
other  x's,  in  the  form 

xi  =  fi(x3,  xt,  .  .  .  ,  xq), 

xz  =  /2(z3,  $«,...,  xq). 

Using  these  in  the  observation  equations  (53)  we  obtain  a  new  set 
of  equations,  independent  of  x\  and  x*t  that  may  be  solved  as 
above.  It  will  be  readily  seen  that  this  process  can  be  extended 
to  include  any  number  of  equations  of  condition. 

When  the  number  of  conditions  is  greater  than  two,  the  compu- 
tation by  the  above  method  becomes  too  complicated  for  practical 
application  and  special  methods  have  been  devised  for  dealing 
with  such  cases.  The  development  of  these  methods  is  beyond 
the  scope  of  the  present  work,  but  they  may  be  found  in  treatises 
on  geodesy  and  practical  astronomy  in  connection  with  the  prob- 
lems to  which  they  apply. 


CHAPTER  VIII. 
PROPAGATION   OF  ERRORS. 

57.  Derived  Quantities.  —  In  one  class  of  indirect  measure- 
ments, the  desired  numeric  -X"  is  obtained  by  computation  from 
the  numerics  Xi,  Xz,  etc.,  of  a  number  of  directly  measured  mag- 
nitudes, with  the  aid  of  the  known  functional  relation 

X  =  F(X1,Xi,  .  .  .  ,Xq). 

We  have  seen  that  the  most  probable  value  that  we  can  assign  to 
the  numeric  of  a  directly  measured  quantity  is  either  the  arith- 
metical mean  of  a  series  of  observations  of  equal  weight  or  the 
general  mean  of  a  number  of  measurements  of  different  weight. 
Consequently,  if  x\,  Xz,  .  .  •  ,  xq  represent  the  proper  means  of 
the  observations  on  Xi,  X2,  .  .  .  ,  Xq  the  most  probable  value 
x  that  we  can  assign  to  X  is  given  by  the  relation 
x  =  F  (xi,  xz,  .  .  .  ,  xq) 

where  F  has  the  same  form  as  in  the  preceding  equation. 

Obviously,  the  characteristic  errors  of  x  cannot  be  easily  deter- 
mined by  a  direct  application  of  the  methods  discussed  in  Chapters 
V  and  VI,  as  this  would  require  a  separate  computation  of  x  from 
each  of  the  individual  observations  on  which  Xi,  Xz,  etc.,  depend. 
Furthermore,  it  frequently  happens  that  we  do  not  know  the 
original  observations  and  are  thus  obliged  to  base  our  computa- 
tions on  the  given  mean  values,  x\,  Xz,  etc.,  together  with  their 
characteristic  errors. 

Hence  it  becomes  desirable  to  develop  a  process  for  computing 
the  characteristic  errors  of  x  from  the  corresponding  errors  of 
Xij  xz,  etc.  For  this  purpose  we  will  first  discuss  several  simple 
forms  of  the  function  F  and  from  the  results  thus  obtained  we 
will  derive  a  general  process  applicable  to  any  form  of  function. 

58.  Errors  of  the  Function  Xi  ±  Xz  ±  X3  =t  .  .  .  ±Xq. 
Suppose  that  the  given  function  is  in  the  form 

X  =  Xi  +  X2,  or  X  =  Xi  -  X2. 

These  two  cases  can  be  treated  together  by  writing  the  function  in 
the  form 

X  =  X\  db  Xz, 
95 


96  THE   THEORY  OF  MEASUREMENTS       [ART.  58 

and  remembering  that  the  sign  ±  indicates  two  separate  problems 
rather  than,  as  usual,  an  indefinite  relation  in  a  single  problem. 
If  the  individual  observations  on  Xi  are  represented  by  ai,  a2, 
.  .  .  ,  an,  and  those  on  X2  by  61,  62,  .  .  .  ,  bn,  we  have 


n  n 

and  the  most  probable  value  of  X  is  given  by  the  relation 

x  =  Xi  ±  xz. 

From  the  given  observations  we  can  calculate  n  independent 
values  of  X  as  follows  : 

Ai  =  ai  ±  &i,       A2  =  az  d=  62,    .  .  .  ,     An  =  aw  db  6n, 

and  it  is  obvious  that  the  mean  of  these  is  equal  to  x.  The  true 
accidental  errors  of  the  a's  are 

Aai  =  oi  —  Xi,  Aa2  =  az  —  Xi,  .  .  .  ,  Aan  =  an  —  Zi; 
those  of  the  6's  are 

Ah  =  61  -  Z2,  A62  =  62  -  Z2,  .  .  .  ,  A6n  =  bn  -  X2; 
and  those  of  the  A's  are 

^Al=A1-X)     &A2=A2-X,    .  .  .  ,  &An=An-X. 

We  cannot  determine  these  errors  in  practice,  since  we  do  not 
know  the  true  value  of  the  X's,  but  we  can  assume  them  in  literal 
form  as  above  for  the  purpose  of  finding  the  relation  between  the 
characteristic  errors  of  the  x's. 

Combining  the  equations  of  the  preceding  paragraph  with  the 
given  functional  relation,  we  have 

AAX=  (ai  ±  60  -  (Zi  ±  Z2) 

=  (a!  -  ZO  ±  (61  -  Xz) 
=  Aai  ±  A&i, 

and  similar  expressions  for  the  other  A  A's.     Consequently 

(AAO2  =  (AaO2  d=  2  AaiA&i  +  (A6i)2, 
(AA2)2  =  (Aa2)2  d=  2  Aa2A62 


(AAn)2  =  (Aan)2  ±  2  kantU)n 
Adding  these  equations,  we  find 

[(AA)2]  =  [(Aa)2]  ±  2  [AaA6]  +  [(A6)2]. 


ART.  58]  PROPAGATION  OF  ERRORS  97 

Since  A  a  and  A  b  are  true  accidental  errors,  they  are  distributed 
in  conformity  with  the  three  axioms  stated  in  article  twenty-four. 
Consequently  equal  positive  and  negative  values  of  Aa  and  A6 
are  equally  probable  and  the  term  [AaA6]  would  vanish  if  an 
infinite  number  of  observations  were  considered.  In  any  case  it 
is  negligible  in  comparison  with  the  other  terms  in  the  above 
equation.  Hence,  on  dividing  through  by  n,  we  have 

[(AA)1  =  [(Aa)«l      [(A6)*]_ 
n  n  n 

and  by  equation  (20),  article  thirty-seven,  this  becomes 

MA2  =  Ma2  +  Mb2,  (73) 

where  MA  is  the  mean  error  of  a  single  A,  Ma  that  of  a  single  a, 
and  Mb  that  of  a  single  b.  Since  x,  xi,  and  z2  are  the  arithmetical 
means  of  the  A's,  a's,  and  6's,  respectively,  their  respective  mean 
errors,  M ,  MI,  and  M 2,  are  given  by  the  relations 

M  2  M  2  Tlf  i2 

M*  =  ^,     itf-=±,    and    M,  =  ^- 

n  n  n 

in  virtue  of  equations  (29),  article  forty.     Consequently,  by  (73) 
M2  =  Mi2  +  M  22, 


or  M  =  VMi2  +  M22.  (74) 

Since  the  mean  and  probable  errors,  corresponding  to  the  same 
series  of  observations,  are  connected  by  the  constant  relation  (26), 
article  thirty-nine,  we  have  also 

+  Ef,  (75) 


where  E,  EI,  and  Ez  are  the  probable  errors  of  x,  x\,  and  #2, 
respectively. 

It  should  be  noticed  that  the  ambiguous  sign  does  not  appear 
in  the  expressions  for  the  characteristic  errors.  The  square  of 
the  error  of  the  computed  quantity  is  equal  to  the  sum  of  the 
squares  of  the  corresponding  errors  of  the  directly  measured  quan- 
tities; whether  the  sign  in  the  functional  relation  is  positive  or 
negative.  Thus  the  error  of  the  sum  of  two  quantities  is  equal 
to  the  corresponding  error  of  the  difference  of  the  same  two  quan- 
tities. 

Now  suppose  that  the  given  functional  relation  is  in  the  form 
X  =  Xi  d=  X2  ±  Xt. 


98  THE   THEORY  OF  MEASUREMENTS      [ART.  59 

The  most  probable  value  of  X  is  given  by  the  relation 

x  =  xi  ±  xz  ±  x3y 

where  the  notation  has  the  same  meaning  as  in  the  preceding 
case.     Represent  x\  ±  xz  by  xp,  then 

a;  =  xp  =t  z3, 

and,  by  an  obvious  extension  of  the  notation  used  above,  we  have 
MP2  =  Mi2  +  M22, 
Mz   =  MP2  +  M32 

=  Mi2  +  M22  +  M32. 

Passing  to  the  more  general  relation 

X  =  Xi  ±  X2  ±  X3  ±  -  -  -  ±  X,, 

we  have  a;  =  £1  db  #2  ±  x3  ±  •  •  •  ±  zfl, 

and,  by  repeated  application  of  the  above  process, 

M2  =  M        M2      MJ  +  -  -  •  +  M32,  ) 


+  -E- 


Thus  the  square  of  the  error  of  the  algebraic  sum  of  a  series  of 
terms  is  equal  to  the  sum  of  the  squares  of  the  corresponding 
errors  of  the  separate  terms  whatever  the  signs  of  the  given  terms 
may  ba 

59.  Errors  of  the  Function  a\Xi  =t  0:2^2  db  asX3  =b  -  •  •  ±  aqXq. 

Let  the  given  functional  relation  be  in  the  form 

X  = 


where  a\  is  any  positive  or  negative,  integral  or  fractional,  con- 
stant. The  most  probable  value  that  we  can  assign  to  X  on  the 
basis  of  n  equally  good  independent  measurements  of  X  is 

x  =  aiXi, 

where  Xi  is  the  arithmetical  mean  of  the  n  direct  observations 
ai,  a2,  as,  .  .  .  ,  an. 

The  n  independent  values  of  X  obtainable  from  the  given  obser- 
vations are 

AI  —  ami,    Az  —  aids,  .  .  .  ,  An  =  a\an. 
The  accidental  errors  of  the  a's  and  A's  are 

Aai  =  a\  —  Xij     Aa2  =  a2  —  X\t  .  .  .  ,  Aan  =  an  —  X\, 
and 

A4i  =  Ai  -  X,    A^2  =  At-X,    .  .  .  ,  AAn  =  An-X. 


ART.  60]  PROPAGATION   OF  ERRORS  99 

Combining  these  equations  we  find 


and  similar  expressions  for  the  other  AA's.     Consequently 

(AAO2  =  ai2(Aax)2, 
and  [(AA)2]  =  ai»  [(Aa)2]. 

If  M  and  Af  i  are  the  mean  errors  of  x  and  xit  respectively, 

and    Jf,..I3. 


Hence                                  M2  =  onWi2,  (77) 

and,  since  the  probable  error  bears  a  constant  relation  to  the 
mean  error, 

E2  =  a^!2.  (78) 

When  the  given  functional  relation  is  in  the  more  general  form 

X  =  aiXi  =b  0:2^2  =b  0.3X3  ±    •   •   •    =b  otqXqj 

we  have 

x  = 


where  the  a;'s  are  the  most  probable  values  that  can  be  assigned 
to  the  X's  on  the  basis  of  the  given  measurements.  Applying 
(77)  and  (78)  to  each  term  of  this  equation  separately  and  then 
applying  (76)  we  have 


t 
E2    = 


where  the  ATs  and  E's  represent  respectively  the  mean  and  prob- 
able errors  of  the  x's  with  corresponding  subscripts. 

60.   Errors  of  the  Function  F  (Xl}  X2,  .  .  .  ,  Xq). 

We  are  now  in  a  position  to  consider  the  general  functional 
relation 

X  =  F  (Xi,  Xz,  .  .  .  ,  Xq), 

where  F  represents  any  function  of  the  independently  measured 
quantities  Xi,  X2,  etc.  Introducing  the  most  probable  values  of 
the  observed  numerics,  the  most  probable  value  of  the  computed 
numeric  is  given  by  the  relation 

x  =  F  fa,  x2)  .  .  .  ,  Xq).  (80) 

This  expression  may  be  written  in  the  form 

&l),     (Z2-f-52..    .  .  .  ,  (*„  +  «,)!,  0) 


100          THE   THEORY  OF  MEASUREMENTS         [ART.  60 

where  the  I's  represent  arbitrary  constants  and  the.S's  are  small 
corrections  given  by  relations  in  the  form 


Obviously,  the  errors  of  the  5's  are  equal  to  the  errors  of  the  corre- 
sponding x's.  For,  if  Mi,  Ms,  and  MI  are  the  errors  of  Xi,  5i,  and 
Zi,  respectively,  we  have  by  equation  (74) 

Ms*  =  Mi2  +  Mf. 

But  MI  is  equal  to  zero,  because  I  is  an  arbitrary  quantity  and  any 
value  assigned  to  it  may  be  considered  exact.  Consequently 

Mi2.  (ii) 


Since  the  I's  are  arbitrary,  they  may  be  so  chosen  that  the 
squares  and  higher  powers  of  the-5's  will  be  negligible  in  compari- 
son with  the  8's  themselves.  Hence,  if  the  x's  are  independent, 
(i)  may  be  expanded  by  Taylor's  Theorem  in  the  form 


dF         d    „,  \      ** 

where  —  =  —  F  (z,  z,  .  .  .  ,  x)  =  —  >  • 

and  the  other  differential  coefficients  have  a  similar  significance. 
When  the  observed  values  of  the  x's  are  substituted  in  these 
coefficients,  they  become  known  numerical  constants. 

The  mean  error  of  F  (li,  Z2,  .  .  .  ,  lq)  is  equal  to  zero,  since  it 
is  a  function  of  arbitrary  constants;  and  the  mean  errors  of  the 
5's  are  equal  to  the  mean  errors  of  the  corresponding  x's  by  (ii). 
Consequently,  if  M,  Mi,  M2,  .  .  .  ,  Mq  represent  the  mean  errors 
of  x,  Xi,  xz,  .  .  .  ,  xq,  respectively,  we  have  by  equation  (79) 

/dF  - .  V  ,  fdF  , ,  V  , 

=  F~MI)  +  brr^2)  + 

\dxi       I       \dx2       I  N~~«       ,  ,     . 

(OL) 


where  the  E's  represent  the  probable  errors  of  the  x's  with  corre- 
sponding subscripts. 

Equations  (81)  are  general  expressions  for  the  mean  and  prob- 
able errors  of  derived  quantities  in  terms  of  the  corresponding 
errors  of  the  independent  components.  Generally  x\t  x2,  etc., 


ART.  61]  PROPAGATION   OF   ERRORS  101 

represent  either  the  arithmetical  or  the  general  means  of  series  of 
direct  observations  on  the  corresponding  components,  and  EI,  Ez, 
etc.,  can  be  computed  by  equations  (32)  or  (41).  In  some  cases, 
the  original  observations  are  not  available  but  the  mean  values 
together  with  their  probable  errors  are  given. 
For  the  purpose  of  computing  the  numerical  value  of  the  differ- 

r\Tj1  r\Tj1 

ential  coefficients  -r—  ;  —  >  etc.,  the  given  or  observed  values  of 
oXi    0X2 

the  components  xi}  x2)  etc.,  may  generally  be  rounded  to  three 
significant  figures.  This  greatly  reduces  the  labor  of  computa- 
tion and  does  not  reduce  the  precision  of  the  result,  since  the  E's 
and  M's  are  seldom  given  or  desired  to  more  than  two  significant 
figures. 

61.  Example  Introducing  the  Fractional  Error.  —  The  prac- 
tical application  of  the  foregoing  process  is  illustrated  in  the  follow- 
ing simple  example:  the  volume  V  of  a  right  circular  cylinder  is 
computed  from  measurements  of  the  diameter  D  and  the  length  L, 
and  we  wish  to  determine  the  probable  error  of  the  result.  In 
this  case,  V  corresponds  to  x,  D  to  xi,  L  to  x2)  and  the  functional 
relation  (80)  becomes 


Also,  if  EV,  ED,  and  EL  are  the  probable  errors  of  V,  D,  and  L, 
respectively,  the  second  of  equations  (81)  becomes 


where 

sv 


and 

dV  d  /I  \  1  n2 
-r^F-  =  ^F  \  -7  TTL)  L  ]  =  —-TrD*. 
dL  dL\4  /  4 

Hence 


The  computation  can  be  simplified  by  introducing  the  frac- 

TTT 

tional  error  -^~-     Thus,  dividing  the  above  equation  by 


we  have 


^=4^!  +  ^ 

T7"O  7~^9          I  T  O 


102  THE  THEORY  OF  MEASUREMENTS       [ART.  62 

or,  writing  PV,  PD,  and  PL  for  the  fractional  errors, 
Py2  =  4  Pz>2  +  PL\ 
PV 
and  finally 

Ev  =  FPF  =  V 


A  similar  simplification  can  be  effected,  in  dealing  with  many 
other  practical  problems,  by  the  introduction  of  the  fractional 
errors.  Consequently  it  is  generally  worth  while  to  try  this  ex- 
pedient before  attempting  the  direct  reduction  of  the  general 
equation  (81).- 

In  order  to  render  the  problem  specific,  suppose  that 
D  =  15.67  ±  0.13  mm., 
L  =  56.25  d=  0.65  mm., 
then  V  =  10848 


PD  =        =          = 

PL  =  ^  =  ^  =  .0116;    Pz,2  =  135  X  10-6, 

=  0.020, 


Ev  =  VTV  =  220  mm 
Hence 

7=  10.85  ±  0.22  cln.3 
62.  Fractional  Error  of  the  Function  aX^1  X  Z2±U2X 


Xan5.- 

Suppose  the  given  relation  is  in  the  form 
X  =  F(Xl)  =aXi± 


where  a  and  n  are  constants  and  the  =fc  sign  of  the  exponent  n  is 
used  for  the  purpose  of  including  the  two  functions  aXi+n  and 
aX-r^  in  the  same  discussion.  In  this  case  equation  (80)  becomes 


x  =  axi±n, 


and  the  second  of  (81)  reduces  to 
But 


_=_ 

Consequently 


ART.  62]  PROPAGATION  OF  ERRORS  103 

If  P  and  PI  are  the  fractional  errors  of  x  and  xi,  respectively,  we 
have 

E* 

- 


Hence 

i  P  =  nP,.  (82) 

If  we  replace  n  by  —  in  the  above  argument,  (80)  becomes 


_ 

x  =  aXi±m, 


and  we  find 


m 

Hence  the  fractional  error  of  any  integral  or  fractional  power  of 
a  measured  numeric  is  equal  to  the  fractional  error  of  the  given 
numeric  multiplied  by  the  exponent  of  the  power. 

If  the  given  function  is  in  the  form  of  a  continuous  product 

X  =  aXl  X  X,  X  •  •  •  X  Xqt 
(80)  becomes         x   =  axi   X  x2   X  •  •  •  X  xq. 

dF 

Hence  —  =  axz   X  x3   X  •  •  •  X  xg, 

ox\ 

I  dF       1 

and  -  —  =  — 

Hence,  by  (81), 

JP  _  Ei2      EJ  Eg2 

rz  ~  7~2  ~f~  ~~2  ~r  T  —£> 

Js  JL>1  JU2  •Lq 

and,  if  P,  PI,  P2,  .  .  .  ,  Pq  represent  the  fractional  errors  of  the 
#'s  with  corresponding  subscripts, 

Combining  the  above  cases  we  obtain  the  more  general  rela- 
tion 

X  =  aXi     1  X  Xz     2  X   •  *  *   X  Xq      , 
and  the  corresponding  expression  for  (80)  is 

Applying  (82)  to  each  factor  separately  and  then  applying  (83)  to 
the  product,  we  find 

f  -  -  -  +nfPf.          (84) 


104  THE   THEORY  OF  MEASUREMENTS       [ART.  62 

For  the  sake  of  illustration  and  to  fix  the  ideas  this  result  may 
be  compared  with  the  example  of  the  preceding  article.     If  we 

put  x  =  V,    Xi  =  D,    HI  =  2,    x2  =  L,    n2  =  1,    a  =  -7  ,    P  =  Py, 
PI  =  PD,  and  PZ  =  PL  the  above  expression  for  x  becomes 

V  =  %TrD2L, 
and  (84)  becomes 


Occasionally  it  is  convenient  to  express  the  probable  error  in 
the  form  of  a  percentage  of  the  measured  magnitude.  If  E  and 
p  are  respectively  the  probable  and  percentage  errors  of  x, 

p=  100  -  =  100  P.  (85) 

x 

Consequently  (84)  may  be  written  in  the  form 

P2  =  niW  +  n22p22  +  •  •  •  +  nfp*,  (84a) 

where  pi,  p2,  .  .  .  ,  pq  are  the  percentage  errors  of  Xi,  x2,  .  .  .  ,  xq, 
respectively 


CHAPTER  IX. 
ERRORS   OF  ADJUSTED   MEASUREMENTS. 

WHEN  the  most  probable  values  of  a  number  of  numerics 
Xi,  X2,etc.,  are  determined  by  the  method  of  least  squares,  the 
results  Xi,  x2,etc.,  are  called  adjusted  measurements  of  the  quan- 
tities represented  by  the  X's.  In  Chapter  VII  we  have  seen  how 
the  x's  come  out  by  the  solution  of  the  normal  equations  (56)  or 
(58),  and  how  these  equations  are  derived  from  the  given  obser- 
vations through  the  equations  (53).  In  the  present  chapter  we 
will  determine  the  characteristic  errors  of  the  computed  x's  in 
terms  of  the  corresponding  errors  of  the  direct  measurements  on 
which  they  depend. 

63.  Weights  of  Adjusted  Measurements.  —  When  there  are  q 
unknowns  and  the  given  observations  are  all  of  the  same  weight, 
the  normal  equations,  derived  in  article  fifty,  are 

[aa]  Xi  +  [ab]  x2  +  [ac]  x3  +  -  •  •  +  [ap]  xq  =  [as], 

[db]  x,  +  [66]  x2  +  [6c]  *,+  •••+  [bp]  xq  =  [bs],  (56) 

[ap]  xi  +  [bp]  xz  +  [cp]  x3  +  •  •  •  +  [pp]  xq  =  [ps]. 

Since  these  equations  are  independent,  the  resulting  values  of  the 
x's  will  be  the  same  whatever  method  of  solution  is  adopted.  In 
Chapter  VII  Gauss's  method  of  substitution  was  used  on  account 
of  the  numerous  checks  it  provides.  For  our  present  purpose 
the  method  of  indeterminate  multipliers  is  more  convenient  as  it 
gives  us  a  direct  expression  for  the  x's  in  terms  of  the  measured 
s's.  Obviously  this  change  of  method  cannot  affect  the  errors  of 
the  computed  quantities. 

Multiply  each  of  equations  (56)  in  order  by  one  of  the  arbitrary 
quantities  AI,  A2,  .  .  .  ,  Aq  and  add  the  products.  The  result- 
ing equation  is 


(86) 


+  ([db]  A1  +  [bb]  A,  +  •  •  •  +  [bp]  Aq)  x2 

+  •        •        •  •      >  •        • 


=    [as]  Al  +  [6s]  A2  +  •  •  •  +  [ps]  Aq. 
105 


[ob]  A,  +  [66]  A*  +  •  •  •  +  [6p]  Aq  =  0, 


106  THE  THEORY  OF  MEASUREMENTS       [ART.  63 

Since  the  A's  are  arbitrary  and  q  in  number,  they  can  be  made  to 
satisfy  any  q  relations  we  choose  without  affecting  the  validity 
of  equation  (86).  Hence,  if  we  determine  the  A's  in  terms  of  the 
coefficients  in  (56)  by  the  relations 


(g7) 


equation  (86)  gives  an  expression  for  x\  in  the  form 

xi  =  [as]  Ai  +  [&*]  4i  +••!-•'+  \ps]At.  (88) 

If  we  repeat  this  process  q  times,  using  a  different  set  of  multipliers 
each  time,  we  obtain  q  different  equations  in  the  form  of  (86). 
In  each  of  these  equations  we  may  place  the  coefficient  of  one  of 
the  x's  equal  to  unity  and  the  other  coefficients  equal  to  zero,  giv- 
ing q  sets  of  equations  in  the  form  of  (87)  for  determining  the  q  sets 
of  multipliers.  Representing  the  successive  sets  of  multipliers  by 
A's,  B's,  C"s,  etc.,  we  obtain  (88),  and  the  following  expressions 
for  the  other  x's  : 

x2  =  [as]  Bi  +  [bs]  ft  +...;+  \p8]  Bq, 
x3  =  [as]  Ci  +  [6s]  C2  +  •  •  •  +  \ps]  Cq, 


xq  =  [as]  P!  +  [6s]  P2  +  •  •  •  +  \ps]  Pq. 

From  equations  (87),  it  is  obvious  that  the  A's  do  not  involve 
the  observations  Si,  s2,  etc.  Consequently  (88)  may  be  expanded 
in  terms  of  the  observations  as  follows: 

Xi  =  ctiSi  +  azs2  -f  •  •  •   +  ctgSq,  (89) 

where  the  a's  depend  only  on  the  coefficients  in  the  observation 

equations  (53)  and  are  independent  of  the  s'a.     Since  we  are  con- 

sidering the  case  of  observations  of  equal  weight,  each  of  the  s's 

in  (89)  is  subject  to  the  same  mean  error  M8.     Her   e,  if  MI  is 

the  mean  error  of  Xi,  we  have  by  equations  (79),  article  fifty-nine, 

Mx2  =  ai2Ms2  +  «22Ma2  +  -  •  •  +  an2M,2 

=  M  M,2. 

But,  if  Wi  is  the  weight  of  x\  in  comparison  with  that  of  a  single  s, 
we  have  by  (36),  article  forty-four, 

Wl   w    i  (90) 

Mi2       [act] 


ART.  63]    ERRORS  OF  ADJUSTED  MEASUREMENTS     107 


since  the  ratio  of  the  mean  errors  of  two  quantities  is  equal  to  the 
ratio  of  their  probable  errors. 

Comparing  equations  (88)  and  (89),  with  the  aid  of  equations 
(55),  article  fifty,  we  see  that 

biA2  +  •  •  •  +piAq, 

(i) 


an  = 


pnAq. 


Multiply  each  of  these  equations  by  its  a  and  add  the  products, 
then  multiply  each  by  its  b  and  add,  and  so  on  until  all  of  the 
coefficients  have  been  used  as  multipliers.  We  thus  obtain  the 
q  sums  [aa],  [ba],  .  .  .  ,  [pa],  and  by  taking  account  of  equations 
(87)  we  have 

[aa]  =  1,  > 

[ba]  =  [ca]  =    •  •  .    =  [pa]  =  0.    ) 

Hence,  if  we  multiply  each  of  equations  (i)  by  its  a  and  add  the 

products,  we  have 

[aa]  =  A  i. 

Consequently  equation  (90)  becomes 


Al 


(91) 


The  weights  of  the  other  x's  may  be  obtained,  by  an  exactly 
similar  process,  from  equations  (88a).  The  results  of  such  an 
analysis  are  as  follows: 


M 


Ma2      Pt 


(91a) 


Obviously  the  coefficients  of  the  sums  [as],  [bs],  etc.,  in  equa- 
tions (88)  and  (88a)  do  not  depend  upon  the  particular  method  by 
which  the  normal  equations  are  solved,  since  the  resulting  values 
of  the  x's  must  be  the  same  whatever  method  is  used.  Conse- 
quently, if  the  absolute  terms  [as],  [bs],  .  .  . ,  [ps]  are  kept  in  literal 
form  during  the  solution  of  the  normal  equations  by  any  method 
whatever,  the  results  may  be  written  in  the  form  of  equations 


108  THE  THEORY  OF  MEASUREMENTS       [ART.  64 

(88)  and  (88a);  and  the  quantities  AI,  B2)  etc.,  will  be  numerical 
if  the  coefficients  [aa],  [ab],  .  .  .  ,  [bb],  .  .  .  ,  [pp]  are  expressed 
numerically. 

Hence,  in  virtue  of  (91)  and  (91  a),  we  have  the  following  rule 
for  computing  the  weights  of  the  z's. 

Retain  the  absolute  terms  of  the  normal  equations  in  literal 
form,  solve  by  any  convenient  method,  and  write  out  the  solution 
in  the  form 

a?i  =  [as]  A!  +  [bs]  A2  +  [cs]  A3  +  -  •  •  +  \ps]  Aqt 
x2  =  [as]  Bl  +  [bs]  B2  +  [cs]  B3  +  -  -  -  +  \ps]  Bq, 

xq  =  [as]  P1  +  [bs]  P2  +  [cs]  P,  +  -  •  -   +  [ps]  Pq. 

Then  the  weight  of  x\  is  the  reciprocal  of  the  coefficient  of  [as]  in 
the  equation  for  x\,  the  weight  of  x2  is  the  reciprocal  of  the  co- 
efficient of  [bs]  in  the  equation  for  x%,  and  in  general  the  weight  of 
xq  is  the  reciprocal  of  the  coefficient  of  [ps]  in  the  equation  for  xq. 

As  an  aid  to  the  memory,  it  may  be  noticed  that  the  coefficients 
AI,  B2,  Cs,  .  .  .  ,  Pq,  that  determine  the  weights,  all  lie  in  the 
main  diagonal  of  the  second  members  of  the  above  equations. 
When  the  number  of  unknowns  is  greater  than  two,  the  labor  of 
computing  all  of  the  A's,  B's,  etc.,  would  be  excessive,  and  conse- 
quently it  is  better  to  determine  the  x's  by  the  methods  of  Chap- 
ter VII.  The  essential  coefficients  AI,  B2,  C3,  .  .  .  ,  Pq  can  be 
determined  independently  of  the  others  by  the  method  of  deter- 
minants as  will  be  explained  later. 

If  the  given  observations  are  not  of  equal  weight,  the  weights 
of  the  x's  may  be  determined  by  a  process  similar  to  the  above, 
starting  with  normal  equations  in  the  form  of  (58),  article  fifty. 
The  result  of  such  an  analysis  can  be  expressed  by  the  rule  stated 
above  if  we  replace  the  sums  [as],  [bs],  .  .  .  ,  [ps]  by  the  weighted 
sums  [was],  [wbs],  .  .  .  ,  [wps],  the  notation  being  the  same  as  in 
article  fifty. 

64.  Probable  Error  of  a  Single  Observation.  —  By  definition, 
article  thirty-seven,  the  mean  error  M8  of  a  single  observation  is 
given  by  the  expression 

_  Af  +  A^+.-.+A.'  _  [AA] ,  (iii) 

n  n 

where  the  A's  represent  the  true  accidental  errors  of  the  s's. 
When  the  number  of  observations  is  very  great,  the  residuals  given 


ART.  64]    ERRORS  OF  ADJUSTED  MEASUREMENTS     109 

by  equations  (54)  may  be  used  in  place  of  the  A's  without  causing 
appreciable  error  in  the  computed  value  of  M8.  But,  in  most 
practical  cases,  n  is  so  small  that  this  simplification  is  not  admis- 
sible and  it  becomes  necessary  to  take  account  of  the  difference 
between  the  residuals  and  the  accidental  errors. 

Let  Ui,  u2,  .  .  .  ,  uq  represent  the  true  errors  of  the  x's  ob- 
tained by  solution  of  the  normal  equations  (56).  Then  the  true 
accidental  error  of  the  first  observation  is  given  by  the  relation 

Ol  (Xi  +  Ui)  +  61  (X2  +  U2)  +    •    •    •     +  Pl  (Xq  +  Uq)  -  Si  =  Ai. 

But,  by  the  first  of  equations  (54), 

aiXi  +  6ix2  -f  cixs  +  •  •  •  +  pixq  —  si  =  ri, 

where  r\  is  the  residual  corresponding  to  the  first  observation. 
Combining  these  equations  and  applying  them  in  succession  to 
the  several  observations,  we  obtain  the  following  expressions  for 
the  A's  in  terms  of  the  r's: 

ri  +  aiui  +  biu2  +  CiU3  +  -  •  •  +  piUq  =  Ai, 

A2, 


,.* 

+  bnu2  +  cnu3  +  •  •  •  +  pnuq  =  An. 
Multiply  each  of  these  equations  by  its  r  and  add;  the  result  is 

[rr]  +  [ar]  HI  +  [br]  u2  +  [cr]  u3  +  •  •  •  +  [pr]  uq  =  [Ar]. 
But  by  equations  (iii),  article  fifty, 

[ar]  =  [br]  =  [cr]  =  •  •  •  =  for]  =  0,  (v) 

and,  consequently, 

[rr]  =  [Ar].  (vi) 

Multiply  each  of  equations  (iv)  by  its  A  and  add.     Then,  taking 
account  of  (vi),  we  have 

[rr]  +  [aA]  Ul  +  [6A]  u2  +  •  •  •  +  [pA]  uq  =  [AA].         (vii) 

In  order  to  obtain  an  expression  for  the  u's  in  terms  of  the  A's, 
multiply  each  of  equations  (iv)  by  its  a  and  add,  then  multiply 
by  the  b's  in  order  and  add,  and  so  on  with  the  other  coefficients. 
The  first  term  in  each  of  these  sums  vanishes  in  virtue  of  (v),  and 
we  have 

[aa]  ui  +  [ab]  w2  +  •  •  •  +  [ap]  uq  =  [aA], 

[db]  Ul  +  [bb]  u,  +  •  •  •  +  \bp]  uq  =  [6A], 


lap]  ui  +  [bp]  u2  +  -  -  -  +  [pp]  uq  = 


(viii) 


110  THE   THEORY  OF  MEASUREMENTS       [Am.  64 

These  equations  are  in  the  same  form  as  the  normal  equations  (56) 
with  the  z's  replaced  by  u's  and  the  s's  by  A's.  Hence  any  solu- 
tion of  (56)  for  the  x's  may  be  transposed  into  a  solution  of  (viii) 
for  the  u's  by  replacing  the  s's  by  A's  without  changing  the  coeffi- 
cients of  the  s's.  Consequently,  by  (89),  we  have 


and  similar  expressions  for  the  other  u's. 
The  coefficients  of  the  u's  in  (vii)  expand  in  the  form 

[aA]  =  aiAi  +  a2A2  +  •  •  •  +  anAn. 
Hence 

[aA]  ui  =  aiaiAi2  +  a2«2A22  +.•••+  ananAn2, 


Since  positive  and  negative  A's  are  equally  likely  to  occur,  the 
sum  of  the  terms  involving  products  of  A's  with  different  subscripts 
will  be  negligible  in  comparison  with  the  other  terms.  The  sum 
of  the  remaining  terms  cannot  be  exactly  evaluated,  but  a  suffi- 
ciently close  approximation  is  obtained  by  placing  each  of  the  A2's 

equal  to  the  mean  square  of  all  of  them,  -  -  -*  Consequently,  as 
the  best  approximation  that  we  can  make,  we  may  put 


n 
But,  by  equations  (ii),  [aa]  is  equal  to  unity.     Hence 

[aA]  „  -  M. 

iv 

Since  there  is  nothing  in  the  foregoing  argument  that  depends  on 
the  particular  u  chosen,  the  same  result  would  have  been  obtained 
with  any  other  u.  .Consequently,  in  equation  (vii),  each  term  that 

involves  one  of  the  u's  must  be  equal  to  -  -  !i  and,  since  there 

tv 

are  q  such  terms,  the  equation  becomes 


Hence,  by  equation  (iii), 
and 


ART.  64]    ERRORS  OF  ADJUSTED  MEASUREMENTS     111 

where  the  r's  represent  the  residuals,  computed  by  equations  (54)  ; 
n  is  the  number  of  observations  ;  and  q  is  the  number  of  unknowns 
involved  in  the  observation  equations  (53).  In  the  case  of  direct 
measurements,  the  number  of  unknowns  is  one,  and  (92)  reduces 
to  the  form  already  found  in  article  forty-one,  equation  (30),  for 
the  mean  error  of  a  single  observation. 

When  the  observations  are  not  of  equal  weight,  the  mean  error 
M8  of  a  standard  observation,  i.e.  an  observation  of  weight 
unity,  is  given  by  the  expression 


2  = 


n 


where  the  w's  are  the  weights  of  the  individual  observations. 
Starting  with  this  relation  in  place  of  (iii)  and  making  correspond- 
ing changes  in  other  equations,  an  analysis  essentially  like  the 
preceding  leads  to  the  result 


Ma  =  ^±'^-,  (93) 

T  n  —  q 

which  reduces  to  the  same  form  as  (92)  when  the  weights  are  all 
unity. 

Introducing  the  constant  relation  between  the  mean  and  probable 
errors,  we  have  the  expressions 

E8  =  0.6741/-M-  ,  (94) 

V  n  —  q 

for  the  probable  error  of  a  single  observation  in  the  case  of  equal 
weights,  and 

E8  =  0.674\/-^i,  (95) 

V  n  —  q 

for  the  probable  error  of  a  standard  observation  in  the  case  of 
different  weights. 

Finally,  if  Mk,  Ek,  and  wk  represent  the  mean  error,  the  probable 
error,  and  the  weight  of  xk,  any  one  of  the  unknown  quantities, 
we  may  derive  the  following  relations  from  the  above  equations 
by  applying  equations  (36),  article  forty-four: 

Ms  

-  =  — 7=    V  ' 

A/in.    T  n  —  o 

(96) 


112 


THE   THEORY  OF  MEASUREMENTS       [ART.  65 


when  the  weights  of  the  given  observations  are  equal,  and 

Mk  =  -^=  =  — L  Y/-^-> 

v  Wk       vWk       n  ~  Q 

„         E,        0.674 

Ek  =     / — -  = 


(97) 


~  2 
when  the  weights  of  the  given  observations  are  not  equal. 

65 .  Application  to  Problems  Involving  Two  Unknowns . — When 
the  observation  equations  involve  only  two  unknown  quantities, 
the  solution  of  the  normal  equations  is  given  by  (59),  article 
fifty-one,  in  the  form 

_  [66]  [as]  -  [ab]  [bs] 
[aa]  [bb]  -  [ab]2   ' 
_  [aa]  [bs]  —  [ab]  [as] 

[aa]  [bb]  -  [ab]2 

By  the  rule  of  article  sixty-three,  the  weight  of  Xi  is  equal  to  the 
reciprocal  of  the  coefficient  of  [as]  in  the  equation  for  Xi,  and  the 
weight  of  #2  is  equal  to  the  reciprocal  of  the  coefficient  of  [bs]  in 
the  equation  for  x2.  Hence,  by  inspection  of  the  above  equations, 
we  have 

[aa]  [bb]  -  [ab]2 


_ 


W2  = 


[bb] 

[aa]  [bb]  -  [ab]2 
[aa] 


(98) 


Since  there  are  only  two  unknown  quantities,  and  the  observa- 
tions are  of  equal  weight,  equation  (92)  gives  the  mean  error  of  a 
single  observation  when  q  is  taken  equal  to  two.  Hence 

(99) 

where  n  is  the  number  of  observation  equations  and  [rr]  is  the 
sum  of  the  squares  of  the  residuals  that  are  obtained  when  the 
computed  values  of  Xi  and  Xz  are  substituted  in  equations  (53a), 
article  fifty-one. 

Combining  equations  (98)  and  (99)  with  (96),  we  obtain  the 
following  expressions  for  the  probable  errors  of  Xi  and  x2: 


0.674 


E2  =  0.674 


v/ 
v/ 


[66] 


[aa][bb]  -  [ab]2    n-2 


[aa] 


[rr> 


[aa]  [bb]  -  [ab]2    n-2 


(100) 


ART.  65]    ERRORS  OF  ADJUSTED  MEASUREMENTS     113 

For  the  purpose  of  illustration,  we  will  compute  the  probable 
errors  of  the  values  of  x\  and  x2  obtained  in  the  numerical  prob- 
lem worked  out  in  article  fifty-one.  Referring  to  the  numerical 
tables  in  that  article,  we  find 

[aa]  =  5;     [ab]  =  20;     [bb]  =  90;    n  =  5; 
[rr]  =  9.60  X  1Q-4. 

Hence,  by  equations  (100), 


*' 


V/ 


5X90-400 

By  equations  (vi),  article  fifty-one,  the  length  L0  of  the  bar  at 
0°  C.,  and  the  coefficient  of  linear  expansion  a  are  given  by  the 
relations 

L0  =  iooo  +  si;   a  =  -L.*». 

10      -L70 

Since  L0  is  equal  to  #1  plus  a  constant,  its  probable  error  is  equal 
to  that  of  Xi  by  the  argument  underlying  equation  (ii),  article 
sixty.  Hence 

EL.  =  E!  =  =fc  0.016. 

To  find  the  probable  error  of  a,  we  have  by  equations  (81),  article 
sixty, 


But,  since  L0  is  very  large  in  comparison  with  x2,  the  second  term 
on  the  right-hand  side  is  negligible  in  comparison  with  the  first. 
Consequently,  without  affecting  the  second  significant  figure  of 
the  result,  we  may  put 


=  Ei  X  10-4  =  =fc  0.038  X  10-5. 

Hence  the  final  results  of  the  computations  in  article  fifty-one  may 
be  more  comprehensively  expressed  in  the  form 

LQ  =  1000.008  db  0.016  millimeters, 
a  =  (1.780  db  0.038)  X  10~5, 


114 


THE  THEORY  OF  MEASUREMENTS       [AET.  66 


when  we  wish  to  indicate  the  precision  of  the  observations  on 
which  they  depend. 
66.  Application  to  Problems  Involving  Three  Unknowns.  —  The 

normal  equations,  for  the  determination  of  three  unknowns,  take 
the  form 

[aa]  Xi  +  [ah]  x2  +  [ac]  x3  =  [as], 


[ac]  xi  +  [be]  x2  +  [cc]  x3  =  [cs]. 
Solving  by  the  method  of  determinants  and  putting 


we  have 


[as] 


x2  =  [as 


[as] 


Hence,  by  the  rule  of  article  sixty-three, 

D 

Wl      [bb][cc]  -[be]2' 

=  D 

2  ~~  [aa]  [cc]  —  [ac]2 ' 

D 
[aa][bb]-[ab]*' 


[aa] 
[ab] 
[ac] 

[ab]     [ac 
[66]     [be 
[be]     [cc 

] 

=  A 

[bb]     [be] 
[be]     [cc] 

1  J 

[be]     [cc] 
[06]     [ac] 

4 

-[cs] 

[06]     [ac] 
[bb]     [be] 

t 

D 

D 

D 

[ac]     [cc] 
[06]     [6c] 

-  +  [6s] 

[aa]    [ac] 
[ac]     [cc] 

- 

-[cs] 

[ab]     [be] 
[aa]     [ac] 

, 

D 

D 

D 

[ah]     [66] 
[ac]     [6c] 

+  N- 

[ac]     [be] 
[aa]    [ab] 

+  [cs] 

[aa]    [ab] 
[ab]     [bb] 

D 

D 

D 

ws  = 


(ix) 


(x) 


The  determinant  D  can  be  eliminated  from  equations  (x),  if 
we  can  obtain  an  independent  expression  for  any  one  of  the  w's. 
The  solution  of  the  normal  equations  by  Gauss's  Method  in 
article  fifty-four  led  to  the  result 


- 
X3~ 


[cc'2] 


ART.  66]    ERRORS  OF  ADJUSTED  MEASUREMENTS     115 

The  auxiliary  [cc  •  2]  is  independent  of  the  absolute  terms  [as], 
[6s],  and  [cs].     The  auxiliary  [cs  •  2]  may  be  expanded  as  follows: 


[oc]r    ,      [6c«l]  („  ,      [ab] 


[6c«l]  („  , 
~  PTTJ  \  M  - 


Hence  the  coefficient  of  [cs]  in  the  above  expression  for  x$  is 
r  -  ~y,  and,  consequently,  the  weight  of  x$  is  equal  to  [cc«2]. 

[CC  •  ZJ 

Substituting  this  value  for  ws  in  the  third  of  equations  (x)  and 
eliminating  D  from  the  other  two  we  have 

[aa]  [bb  •  1] 


[66 


(101) 


w3  =  [cc  •  2], 

where  the  auxiliary  quantities  [66  •  1],  [cc»  1],  and  [cc  •  2]  have  the 
same  significance  as  in  article  fifty-four. 

The  weights  of  the  x's  having  been  determined  by  equations 
(101),  their  probable  errors  may  be  computed  by  equations  (96). 
In  the  present  case  q  is  taken  equal  to  three,  since  there  are  three 
unknowns,  and  the  r's  are  given  by  equations  (68). 

In  the  numerical  illustration  of  Gauss's  Method,  worked  out  in 
article  fifty-five,  we  found  the  following  values  of  the  quantities 
appearing  in  equations  (96)  and  (101): 

[aa]  =  6;   [66]  =  220;   [6c]  =  180;   [cc]  =  157; 
[66  •  1]  =  70;   [cc  •  1]  =  76.0;    [cc  •  2]  =  5.97; 
[rr]  =  0.00120;   n  =  6;   q  =  3. 

These  values  have  been  rounded  to  three  significant  figures,  when 
necessary,  since  the  probable  errors  of  the  #'s  are  desired  to  only 
two  significant  figures.  Substituting  in  equations  (101)  we  have 

Wl  = 6X7°_2  5.97  -1.17, 

220  X  157  -  180 

70 
^2  =  y^5.97  =  5.50, 

w3  =  5.97, 


116  THE  THEORY  OF  MEASUREMENTS       [ART.  66 


From  equation  (94) 

\E. 

and,  by  equations  (96), 


a  =  0.674 1/0'0012  =  ±  0.0135, 


0.0135 
.         —  ±  O.UUoo. 


Consequently  the  precision  of  the  measurements,  so  far  as  it 
depends  on  accidental  errors,  may  be  expressed  by  writing  the 
computed  values  of  the  x's  in  the  form 

xi  =  0.245  ±  0.012, 
X2=-  1.0003  ±  0.0057, 
z3  =  1.4022  ±0.0055. 

Since  the  last  significant  figure  in  each  of  the  x's  occupies  the  same 
place  as  the  second  significant  figure  in  the  corresponding  prob- 
able error,  it  is  evident  that  the  proper  number  of  figures  were 
retained  throughout  the  computations  in  article  fifty-five. 


CHAPTER  X. 
DISCUSSION   OF   COMPLETED   OBSERVATIONS. 

67.  Removal  of  Constant  Errors.  —  The  discussion  of  acci- 
dental errors  and  the  determination  of  their  effect  on  the  result 
computed  from  a  given  series  of  observations,  as  carried  out  in  the 
preceding  chapters,  are  based  on  the  assumption  that  the  meas- 
urements are  entirely  free  from  constant  errors  and  mistakes. 
Hence  the  first  matter  of  importance,  in  undertaking  the  reduction 
of  observations,  is  the  determination  and  removal  of  all  constant 
errors  and  mistakes.  Also,  in  criticizing  published  or  reported 
results,  judgment  is  based  very  largely  on  the  skill  and  care  with 
which  such  errors  have  been  treated.  In  the  former  case,  if  suit- 
able methods  and  apparatus  have  been  chosen  and  the  adjust- 
ments of  instruments  have  been  properly  made,  sufficient  data  is 
usually  at  hand  for  determining  the  necessary  corrections  within 
the  accidental  errors.  In  the  latter  case  we  must  rely  on  the  dis- 
cussion of  methods,  apparatus,  and  adjustments  given  by  the 
author  and  very  little  weight  should  be  given  to  the  reported 
measurements  if  this  discussion  is  not  clear  and 'adequate. 

No  evidence  can  be  obtained  from  the  observations  themselves 
regarding  the  presence  or  absence  of  strictly  constant  errors. 
The  majority  of  them  are  due  to  inexact  graduation  of  scales, 
imperfect  adjustment  of  instruments,  personal  peculiarities  of  the 
observer,  and  faulty  methods  of  manipulation.  They  affect  all 
of  the  observations  by  the  same  relative  amount.  Their  detec- 
tion and  correction  or  elimination  depend  entirely  on  the  judg- 
ment, experience,  and  care  of  the  observer  and  the  computer. 
When  the  same  magnitude  has  been  measured  by  a  number  of 
different  observers,  using  different  methods  and  apparatus,  the 
probability  that  the  constant  errors  have  been  the  same  in  all  of 
the  measurements  is  very  small.  Consequently  if  the  corrected 
results  agree,  within  the  accidental  errors  of  observation,  it  is 
highly  probable  that  they  are  free  from  constant  errors.  This  is 
the  only  criterion  we  have  for  the  absence  of  such  errors  and  it 

117 


118  THE   THEORY  OF  MEASUREMENTS       [ART.  67 

breaks  down  in  some  cases  when  the  measured  magnitude  is  not 
strictly  constant. 

Sometimes  constant  errors  are  not  strictly  constant  but  vary 
progressively  from  observation  to  observation  owing  to  gradual 
changes  in  surrounding  conditions  or  in  the  adjustment  of  instru- 
ments. The  slow  expansion  of  metallic  scales  due  to  the  heat 
radiated  from  the  body  of  the  observer  is  an  illustration  of  a 
progressive  change.  Such  variations  are  usually  called  systematic 
errors.  They  may  be  corrected  or  eliminated  by  the  same  methods 
that  apply  to  strictly  constant  errors  when  adequate  means  are 
provided  for  detecting  them  and  determining  the  magnitude  of 
the  effects  produced.  When  their  range  in  magnitude  is  compara- 
ble with  that  of  the  accidental  errors,  their  presence  can  usually  be 
determined  by  a  critical  study  of  the  given  observations  and  their 
residuals.  But,  if  they  have  not  been  foreseen  and  provided  for 
in  making  the  observations,  their  correction  is  generally  difficult 
if  not  impossible.  In  many  cases  our  only  recourse  is  a  new  series 
of  observations  taken  under  more  favorable  conditions  and  accom- 
panied by  adequate  means  of  evaluating  the  systematic  errors. 

A  general  discussion  of  the  nature  of  constant  errors  and  of  the 
methods  by  which  they  are  eliminated  from  single  direct  observa- 
tions was  given  in  Chapter  III.  These  processes  will  now  be  con- 
sidered a  little  more  in  detail  and  extended  to  the  arithmetical 
mean  of  a  number  of  direct  observations.  Let  a\t  d2,  as,  .  .  .  ,  an 
represent  a  series  of  direct  observations  after  each  one  of  them 
has  been  corrected  for  all  constant  errors.  Then  the  most  prob- 
able value  that  can  be  assigned  to  the  numeric  of  the  measured 
magnitude  is  the  arithmetical  mean 

x  =  qi  +  fl2  +  •  •  •  +an  /jx 

IV 

Now  suppose,  that  the  actual  uncorrected  observations  are  01,  o2, 

o3,  •  •  •  ,  on,  then 

ai  =  01  +  cj  +  cj'  +  cj"  +  •  •  •  +  ci<*>  =  01  +  [cj, 
a2  =  o2  +  cj  +  c2"  +  cj"  +  •  •  •  +  c2(">  =  o2  + 

C*n  =  On  +  Cn'  +  C«"  +  Cn'"  +    •    •    •    +  cj*   =  On+  [c 

where  the  c's  represent  the  constant  errors  to  be  eliminated  and 
may  be  either  positive  or  negative.  There  are  as  many  c's  in 
each  equation  as  there  are  sources  of  constant  error  to  be  consid- 


ART.  67]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  119 

ered.  Usually,  when  all  of  the  observations  are  made  by  the 
same  method  and  with  equal  care,  the  number  of  c's  is  the  same 
in  all  of  the  equations.  Substituting  (ii)  in  (i) 

J.  =  0l  +  02+    •••+«.         [Cj  +  [cj+    •    -    -    +[ftj 

n  n  ' 


When  there  are  no  systematic  errors 

Cl    =  Cz        =  C3'      = 
Cl"    =  C2"     =  C,"     =     •    •    •     =  Cn"    =  C", 


=  C3'      =     *    •    •     =  Cn 


Consequently 

[ci]  =  [cz]  =  [c3]  =  •  •  •  •   =  [cn]  =  [c],  (iv) 

and  we  have 

x  =  — — —  +  [c] 

n 

=  Om  +  c'  +  c"  +  c"'  +  •  •  •  -f  c<«>,          (102) 

where  om  is  written  for  the  mean  of  the  actual  observations. 
Hence,  when  all  of  the  observations  are  affected  by  the  same  con- 
stant errors,  the  corrections  may  be  applied  to  the  arithmetical 
mean  of  the  actual  observations  and  the  resulting  value  of  x  will 
be  the  same  as  if  the  observations  were  separately  corrected  before 
taking  the  mean. 

The  residuals  corresponding  to  the  corrected  observations  ai, 
a2,  a3,  .  .  .  ,  an  are  given  by  equations  (3),  article  twenty-two. 
Replacing  x  and  the  a's  by  their  values  in  terms  of  om  and  the 
o's  as  given  in  (102)  and  (ii),  and  taking  account  of  (iv),  equations 
(3)  become 

ri  =  di  —  X  =  Oi+  [Ci]  -  Om-  [C\   =  01  -  Om, 

r2  =  a2  —  x  =  o2  +  [c2]  —  om  —  [c]  =  o2  —  om,  (103) 

rn  =  an  -  X  =  On  +  [Cn]  -Om-  [c]  =  On  -  Om. 

Consequently,  when  there  are  no  systematic  errors,  the  residuals 
computed  from  the  o's  and  om  will  be  identical  with  those  com- 
puted from  the  a's  and  x.  Hence,  if  the  uncorrected  observations 
are  used  in  computing  the  probable  error  of  x,  by  the  formula 

/      W 

E  =  0.674\/    /    J  1X> 
V  n  (n  —  1) 


120  THE   THEORY  OF  MEASUREMENTS       [ART.  67 

the  result  will  be  the  same  as  if  the  corrected  observations  had 
been  used;  and,  as  pointed  out  above,  the  observations  and  their 
corresponding  residuals  give  no  evidence  of  the  presence  of  strictly 
constant  errors. 

When  the  constant  errors  affecting  the  different  observations 
are  different  or  when  any  of  them  are  systematic  in  character, 
equation  (iv)  no  longer  holds,  and,  consequently,  the  simplifica- 
tion expressed  by  (102)  is  no  longer  possible.  In  the  former  case 
the  observations  should  be  individually  corrected  before  the  mean 
is  taken.  The  same  result  might  be  obtained  from  equation  (iii), 
but  the  computation  would  not  be  simplified  by  its  use.  In  the 
latter  case  the  several  observations  are  affected  by  errors  due  to 
the  same  causes  but  varying  progressively  in  magnitude  in  response 
to  more  or  less  continuous  variations  in  the  conditions  under 
which  they  are  made. 

In  equations  (ii)  the  c's  having  the  same  index  may  be  con- 
sidered to  be  due  to  the  same  cause,  but  to  vary  in  magnitude 
from  equation  to  equation  as  indicated  by  the  subscripts.  The 
arithmetical  means  of  the  errors  due  to  the  same  causes  are 

,    _  Ci'   +  C2'  +     •    •     •     +  Cn' 

Cm   '~  ~ 


_ 

Cm        - 


n 
and  the  mean  of  the  observations  is 

01  +  02  +     '     '     ' 


Om  = 


n 


Substituting  (ii)  in  (i)  and  taking  account  of  the  above  relations 
we  have 

X   =   Om  +  Cm'  +  Cm"  +     '     '     '     +  Cw<«> .  (104) 

Hence,  in  the  case  of  systematic  errors,  the  most  probable  value 
of  the  numeric  of  the  measured  magnitude  may  be  obtained  from 
the  mean  of  the  uncorrected  observations  by  applying  mean  cor- 
rections for  the  systematic  errors.  When  all  of  the  errors  are 
strictly  constant  equation  (104)  becomes  identical  with  (102) 
because  all  of  the  errors  having  the  same  index  are  equal.  Obvi- 


ART.  68]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  121 

ously  it  also  holds  when  part  of  the  c's  are  strictly  constant  and  the 
remainder  are  systematic. 

If  we  use  the  value  of  x  given  by  (104)  in  place  of  that  given 
by  (102)  in  the  residual  equations  (103),  the  c's  will  not  cancel. 
Hence,  if  any  of  the  constant  errors  are  systematic  in  nature,. the 
residuals  computed  from  the  o's  and  om  will  be  different  from 
those  computed  from  the  a's  and  x;  and,  consequently,  they  will 
not  be  distributed  in  accordance  with  the  law  of  accidental  errors. 

In  practice  it  is  generally  advisable  to  correct  each  of  the  ob- 
servations separately  before  taking  the  mean  rather  than  to  use 
equation  (104),  since  the  true  residuals  are  required  in  computing 
the  probable  error  of  x,  and  they  cannot  be  derived  from  the  un- 
corrected  observations.  Whenever  possible  the  conditions  should 
be  so  chosen  that  systematic  errors  are  avoided  and  then  the 
necessary  computation  can  be  made  by  equations  (102)  and  (103). 

68.  Criteria  of  Accidental  Errors.  —  We  have  seen  that  the 
residuals  computed  from  observations  affected  by  systematic  errors 
do  not  follow  the  law  of  accidental  errors.  Hence,  if  it  can  be 
shown  that  the  residuals  computed  from  any  given  series  of  obser- 
vations are  distributed  in  conformity  with  the  law  of  errors,  it  is 
probable  that  the  given  observations  are  free  from  systematic 
errors  or  that  such  errors  are  negligible  in  comparison  with  the 
accidental  errors.  Observations  that  satisfy  this  condition  may 
or  may  not  be  free  from  strictly  constant  errors,  but  necessary 
corrections  can  be  made  by  equation  (102)  and  the  probable  error 
of  the  mean  may  be  computed  from  the  residuals  given  by 
equation  (103). 

Systematic  errors  should  be  very  carefully  guarded  against  in 
making  the  observations,  and  the  conditions  that  produce  them 
should  be  constantly  watched  and  recorded  during  the  progress 
of  the  work.  After  the  observations  have  been  completed  they 
should  be  individually  corrected  for  all  known  systematic  errors 
before  taking  the  mean.  The  strictly  constant  errors  may  then 
be  removed  from  the  mean,  but  before  this  is  done  it  is  well  to 
compute  the  residuals  and  see  if  they  satisfy  the  law  of  accidental 
errors.  If  they  do  not,  search  must  be  made  for  further  causes 
of  systematic  error  in  the  conditions  surrounding  the  measure- 
ments and  a  new  series  of  observations  should  be  made,  under 
more  favorable  conditions,  whenever  sufficient  data  for  this  pur- 
pose is  not  available. 


122  THE  THEORY  OF  MEASUREMENTS       [ART.  68 

Residuals,  when  sufficiently  numerous,  follow  the  same  law  of 
distribution  as  the  true  accidental  errors.  Consequently  system- 
atic errors  and  mistakes  might  be  detected  by  a  direct  comparison 
of  the  actual  distribution  with  the  theoretical,  as  carried  out  in 
article  thirty-four,  provided  the  number  of  observations  is  very 
large.  However,  in  most  practical  measurements,  the  residuals 
are  not  sufficiently  numerous  to  fulfill  the  conditions  underlying 
the  law  of  errors,  and  a  considerable  difference  between  their 
actual  and  theoretical  distribution  is  quite  as  likely  to  be  due  to^ 
this  fact  as  to  the  presence  of  systematic  errors.  Whatever  the 
number  of  observations,  a  close  agreement  between  theory  and 
practice  is  strong  evidence  of  the  absence  of  such  errors  but  it  is 
seldom  worth  while  to  carry  out  the  comparison  with  less  than 
one  hundred  residuals. 

When  the  residuals  are  numerous  and  distributed  in  the  same 
manner  as  the  accidental  errors,  the  average  error  of  a  single 
observation,  computed  by  the  formula 


Vn(n-  1)' 
and  the  mean  error,  computed  by  the  formula 


satisfy  the  relation 

M  =  1.253  A. 
Also  the  formulae 

E  =  0.8453  A     and    E  =  0.6745  M 

give  the  same  value  for  the  probable  error  of  a  single  observation. 
When  the  number  of  observations  is  limited,  exact  fulfillment  of 
these  relations  ought  not  to  be  expected,  but  a  large  deviation 
from  them  is  strong  evidence  of  the  presence  of  systematic  errors 
or  mistakes.  Unless  the  number  of  observations  is  very  small, 
ten  or  less,  the  relations  should  be  fulfilled  within  a  few  units  in 
the  second  significant  figure,  as  is  the  case  in  the  numerical  example 
worked  out  in  article  forty-two. 

Obviously  the  arithmetical  mean  is  independent  of  the  order 
in  which  the  observations  are  arranged  in  taking  it,  but  the  order 
of  the  residuals  in  regard  to  sign  and  magnitude  depends  on  the 
order  of  the  observations.  When  there  are  systematic  errors  and 
the  observations  are  arranged  in  the  order  of  progression  of  their 


ART.  68]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  123 

cause,  the  residuals  will  gradually  increase  or  decrease  in  absolute 
magnitude  in  the  same  order;  and,  if  the  systematic  errors  are 
large  in  comparison  with  the  accidental  errors,  there  will  be  but 
one  change  of  sign  in  the  series.  Thus,  if  the  temperature  is 
gradually  rising  while  a  length  is  being  measured  with  a  metallic 
scale  and  the  observations  are  arranged  in  the  order  in  which  they 
are  taken,  the  first  half  of  them  will  be  larger  than  the  mean  and 
the  last  half  smaller,  except  for  the  variations  caused  by  accidental 
errors.  For  the  purpose  of  illustration,  suppose  that  the  observa- 
tions are 

1001.0;  1000.9;  1000.8;  1000.7;  1000.6;  1000.5;  1000.4. 
The  mean  is  1000.7  and  the  residuals 

+  .3;     +.2;     +.1;     ±.0;      -.1;      -.2;     -.3 

decrease  in  absolute  magnitude  from  left  to  right,  i.e.,  in  the  order 
in  which  the  observations  were  made.  There  are  five  cases  in 
which  the  signs  of  succeeding  residuals  are  alike  and  one  in  which 
they  are  different;  the  former  cases  will  be  called  sign-follows  and 
the^latter  a  sign-change.  This  order  of  the  residuals  in  regard  to 
magnitude  and  sign  is  typical  of  observations  affected  by  sys- 
tematic errors  when  they  are  arranged  in  conformity  with  the 
changes  in  surrounding  conditions.  Since  such  changes  are  usually 
continuous  functions  of  the  time,  the  required  arrangement  is 
generally  the  order  in  which  the  observations  are  taken. 

Such  extreme  cases  as  that  illustrated  above  are  seldom  met 
with  in  practice  owing  to  the  impossibility  of  avoiding  accidental 
errors  of  observation  and  the  complications  they  produce  in  the 
sequence  of  residuals.  Generally  the  systematic  errors  that  are 
not  readily  discovered  and  corrected  before  making  further  re- 
ductions are  comparable  in  magnitude  with  the  accidental  errors. 
Consequently  they  cannot  control  the  sequence  in  the  signs  of 
the  residuals  but  they  do  modify  the  sequence  characteristic  of 
true  accidental  errors. 

In  any  extended  series  of  observations  there  should  be  as  many 
negative  residuals  as  positive  ones,  since  positive  and  negative 
errors  are  equally  likely  to  occur.  After  any  number  of  observations 
have  been  made,  the  probability  that  the  residual  of  the  next  obser- 
vation will  be  positive  is  equal  to  the  probability  that  it  will  be  nega- 
tive, since  the  possible  number  of  either  positive  or  negative  errors 
is  infinite.  Consequently  the  chance  that  succeeding  residuals 


124  THE  THEORY  OF  MEASUREMENTS       [ART.  69 

will  have  the  same  sign  is  equal  to  the  chance  that  they  will  have 
different  signs.  Hence,  if  the  residuals  are  arranged  in  the  order 
in  which  the  corresponding  observations  were  made,  the  number 
of  sign-follows  should  be  equal  to  the  number  of  sign-changes. 

The  residuals,  computed  from  limited  series  of  observations, 
seldom  exhibit  the  theoretical  sequence  of  signs  exactly  because 
they  are  not  sufficiently  numerous  to  fulfill  the  underlying  condi- 
tions. Nevertheless,  a  marked  departure  from  that  sequence 
suggests  the  presence  of  systematic  errors  or  mistakes  and  should 
lead  to  a  careful  scrutiny  of  the  observations  and  the  conditions 
under  which  they  were  made.  If  the  disturbing  causes  cannot  be 
detected  and  their  effects  eliminated,  it  is  generally  advisable  to 
repeat  the  observations  under  more  favorable  conditions.  The 
numerical  example,  worked  out  in  article  forty-two,  may  be  cited 
as  an  illustration  from  practice.  The  observations  were  made  in 
the  order  in  which  they  are  tabulated,  beginning  at  the  top  of  the 
first  column  and  ending  at  the  bottom  of  the  fourth  column.  In 
the  second  and  fifth  columns  we  find  ten  positive  and  ten  negative 
residuals.  The  number  of  sign-follows  is  ten  and  the  number  of 
sign-changes  is  nine.  This  is  rather  better  agreement  with  the 
theoretical  sequence  of  signs  than  is  usually  obtained  with  so  few 
residuals.  It  indicates  that  the  observations  were  made  under 
favorable  conditions  and  are  sensibly  free  from  systematic  errors 
but  it  gives  no  evidence  whatever  that  strictly  constant  errors 
are  absent. 

Although  the  foregoing  criteria  of  accidental  errors  are  only 
approximately  fulfilled  when  the  number  of  observations  is  lim- 
ited, their  application  frequently  leads  to  the  detection  and  elimi- 
nation of  unforeseen  systematic  errors.  The  first  method  is  rather 
tedious  and  of  little  value  when  less  than  one  hundred  obser- 
vations are  considered,  but  the  last  two  methods  may  be  easily 
carried  out  and  are  generally  exact  enough  for  the  detection  of 
systematic  errors  comparable  in  magnitude  with  the  probable  error 
of  a  single  observation. 

69.  Probability  of  Large  Residuals.  —  In  discussing  the  dis- 
tribution of  residuals  in  regard  to  magnitude,  the  words  large  and 
small  are  used  in  a  comparative  sense.  A  large  residual  is  one  that 
is  large  in  comparison  with  the  majority  of  residuals  in  the  series 
considered.  Thus,  a  residual  that  would  be  classed  as  large  in  a 
series  of  very  precise  observations  would  be  considered  small  in 


ART.  69]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  125 

dealing  with  less  exact  observations.  Consequently,  in  expressing 
the  relative  magnitudes  of  residuals,  it  is  customary  to  adopt  a 
unit  that  depends  on  the  precision  of  the  measurements  considered. 
The  probable  error  of  a  single  observation  is  the  best  magnitude 
to  adopt  for  this  purpose,  since  it  is  greater  than  one-half  of  the 
errors  and  less  than  the  other  half.  If  we  represent  the  relative 
magnitude  of  a  given  error  by  S,  the  actual  magnitude  by  A,  and 
the  probable  error  of  a  single  observation  by  E, 

S  =  |-  (105) 

The  relative  magnitudes  of  the  residuals  may  be  represented  in 
the  same  way  by  replacing  the  error  A  by  the  residual  r.  It  is 
obvious  that  values  of  S  less  than  unity  correspond  to  small  re- 
siduals and  values  greater  than  unity  to  large  residuals  in  any 
series  of  observations. 

In  equation  (13),  article  thirty-three,  the  probability  that  an 
error  chosen  at  random  is  less  than  a  given  error  A  is  expressed 

by  the  integral 

*/~  A 
o      /»  v™j 

PA  =  -^=  e-*dt.  (13) 

V-n-Jo 

Equation  (25),  article  thirty-eight,  may  be  put  in  the  form 

V         **       k 

&  =  —7=  •  -> 

VTT     a? 

where  $  is  written  for  the  numerical  constant  0.47694.  Hence, 
introducing  (105), 


and  (13)  becomes 

P8=  'eft.  (106) 


Obviously  this  integral  expresses  the  probability  that  an  error 
chosen  at  random  is  less  than  S  times  the  probable  error  of  a 
single  observation.  It  is  independent  of  the  particular  series  to 
which  the  observations  belong  and  its  values,  corresponding  to 
a  series  of  values  of  the  argument  S,  are  given  in  Table  XII. 

Since  all  of  the  errors  in  any  system  are  less  than  infinity,  Poo 
is  equal  to  unity.     Hence  the  probability  that  a  single  error, 


126 


THE   THEORY  OF  MEASUREMENTS       [ART.  69 


chosen  at  random,  is  greater  than  S  times  E  is  given  by  the  rela- 
tion 

Qs  =   1   -  Pa-  (V) 

Now  the  residuals,  when  sufficiently  numerous  and  free  from 
systematic  errors  and  mistakes,  should  follow  the  same  distri- 
bution as  the  accidental  errors.  Hence,  if  ns  is  the  number  of 
residuals  numerically  greater  than  SE  and  N  is  the  total  number 
in  any  series  of  observations,  we  should  have 

Qs  =  T?"  (vi) 

Since  the  numerical  value  of  P8,  and  consequently  that  of  Q8 
depends  only  on  the  limit  S  and  is  independent  of  the  precision 

of  the  particular  series  of  measurements  considered,  the  ratio  jj.  > 

corresponding  to  any  given  limit  S,  should  be  the  same  in  all 
cases.  Consequently,  if  N  observations  have  been  made  on  any 
magnitude  and  by  any  method  whatever,  n8  of  them  should  corre- 
spond to  residuals  numerically  greater  than  SE.  Conversely,  if 
we  assign  any  arbitrary  number  to  na,  equation  (vi)  defines  the 
number  of  observations  that  we  should  expect  to  make  without 
exceeding  the  assigned  number  of  residuals  greater  than  SE. 
Hence,  if  Na  is  the  number  of  observations  among  which  there 
should  be  only  one  residual  greater  than  S  times  the  probable 
error  of  a  single  observation,  we  have,  by  placing  ns  equal  to 
one  in  (vi),  and  substituting  the  value  of  Q8  from  (v), 

*-£-r^>r  (107) 

The  fourth  column  of  the  following  table  gives  the  values  of  Na, 
to  the  nearest  integer,  corresponding  to  the  integral  values  of  the 
limit  S  given  in  the  first  column.  The  values  of  P8  in  the  second 
column  are  taken  from  Table  XII,  and  those  of  Q8  in  the  third 
column  are  computed  by  equation  (v). 


S 

P. 

e. 

Ns 

1 

0.50000 

0.50000 

2 

2 

0.82266 

0.17734 

6 

3 

0.95698 

0.04302 

23 

4 

0.99302 

0.00698 

143 

5 

0.99926 

0.00074 

1351 

ART.  70]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  127 

To  illustrate  the  significance  of  this  table,  suppose  that  143 
direct  observations  have  been  made  on  any  magnitude  by  any 
method  whatever.  The  probable  error  E  of  a  single  observation 
in  this  series  may  be  computed  from  the  residuals  by  equation  (31) 
or  (34).  Then,  if  the  residuals  follow  the  law  of  errors,  not  more 
than  one  of  them  should  be  greater  than  four  times  as  large  as  E. 
If  the  number  of  observations  had  been  1351,  we  should  expect 
to  find  one  residual  greater  than  five  times  E,  and  on  the  other 
hand  if  the  number  had  been  only  twenty-three,  not  more  than 
one  residual  should  be  greater  than  three  times  E. 

Although  the  probability  for  the  occurrence  of  large  residuals 
is  small,  and  very  few  of  them  should  occur  in  limited  series 
of  observations,  their  distribution  among  the  observations,  in 
respect  to  the  order  in  which  they  occur,  is  entirely  fortuitous. 
A  large  residual  is  as  likely  to  occur  in  the  first,  or  any  other, 
observation  of  an  extended  series  as  in  the  last  observation.  Con- 
sequently the  limited  series  of  observations,  taken  in  practice, 
frequently  contain  abnormally  large  residuals.  This  is  not  due 
to  a  departure  from  the  law  of  errors,  but  to  a  lack  of  sufficient 
observations  to  fulfill  the  theoretical  conditions.  In  such  cases 
there  are  not  enough  observations  with  normal  residuals  to  balance 
those  with  abnormally  large  ones.  Consequently  a  closer  approxi- 
mation to  the  arithmetical  mean  that  would  have  been  obtained 
with  a  more  extended  series  of  observations  is  obtained  when  the 
abnormal  observations  are  rejected  from  the  series  before  taking 
the  mean. 

Observations  should  not  be  rejected  simply  because  they  show 
large  residuals,  unless  it  can  be  shown  that  the  limit  set  by  the 
theory  of  errors,  for  the  number  of  observations  considered,  is 
exceeded.  This  can  be  judged  approximately  by  comparing  the 
residuals  of  the  given  observations  with  the  numbers  given  in  the 
first  and  last  columns  of  the  above  table,  but  a  more  rigorous  test 
is  obtained  by  applying  Chauvenet's  Criterion,  as  explained  in  the 
following  article. 

70.  Chauvenet's  Criterion.  —  The  probability  that  the  error 
of  a  single  observation,  chosen  at  random,  is  less  than  SE  is 
expressed  by  Pa  in  equation  (106).  Now,  the  taking  of  N  inde- 
pendent observations  is  equivalent  to  N  selections  at  random  from 
the  infinite  number  of  possible  accidental  errors.  Hence,  by 
equation  (7),  article  twenty-three,  the  probability  that  each  of 


128  THE   THEORY  OF  MEASUREMENTS       [ART.  70 

the  N  observations  in  any  series  is  affected  by  an  error  less  than 
SE  is  equal  to  P»N.  Since  all  of  the  N  errors  must  be  either  greater 
or  less  than  SE}  the  probability  that  at  least  one  of  them  is  greater 
than  this  limit  is  equal  to  1  —  P8N.  Placing  this  probability 
equal  to  one-half,  we  have 

i  -  P."  =  i, 

or 

P.  -  (1  -  (vii) 


If  the  limit  S  is  determined  by  this  equation,  there  is  an  even 
chance  that  at  least  one  of  the  N  observations  is  affected  by  an 
error  greater  than  SE. 

Expanding  the  second  member  of  (vii)  by  the  Binomial  Theorem 

11       N  -I     I      (N-  l)(2N-l)    1 


N  2      1-2-N2   4  1-2-  3-  N*         8 


1-2-3 .  .  .  K-NK 

The  terms  of  this  series  decrease  very  rapidly  and  all  but  the  first 
are  negative.  Consequently  the  sum  of  the  terms  beyond  the 
second  is  small  in  comparison  with  the  other  two;  and,  whatever 

the  value  of  N,  (1  —  %)N  is  nearly  equal  to,  but  always  slightly 
less  than,  - — ^-^ — - .  Since  P8  and  S  increase  together,  the  limit 
T  determined  by  the  relation 

2N-1 


2N 


(108) 


is  slightly  greater  than  the  limit  S  determined  by  (vii).  Hence, 
if  N  independent  direct  observations  have  been  made,  the  prob- 
ability against  the  occurrence  of  a  single  error  greater  than 

Ar  =  TE  (109) 

is  greater  than  the  probability  for  its  occurrence.  Consequently, 
if  the  given  series  contains  a  residual  greater  than  Ar,  the  prob- 
able precision  of  the  arithmetical  mean  is  increased  by  excluding 
the  corresponding  observation. 


ART.  70]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  129 

Equations  (108)  and  (109)  express  Chauvenet's  Criterion  for  the 
rejection  of  doubtful  observations.  In  applying  them,  the  prob- 
able error  E  of  a  single  observation  is  first  computed  from  the 
residuals  of  all  of  the  observations  by  either  equation  (31)  or  the 
first  of  equations  (34)  with  the  aid  of  Table  XIV  or  XV.  If  any 
of  the  residuals  appear  large  in  comparison  with  the  computed 
value  of  E,  PT  is  determined  from  (108)  by  placing  N  equal  to 
the  number  of  observations  in  the  given  series.  T  is  then  obtained 
by  interpolation  from  Table  XII,  and  finally  Ar  is  computed  by 
(109).  If  one  or  more  of  the  residuals  are  greater  than  the  com- 
puted Ar,  the  observation  corresponding  to  the  largest  of  them  is 
excluded  from  the  series  and  the  process  of  applying  the  criterion  is 
repeated  from  the  beginning.  If  one  or  more  of  the  new  residuals 
are  greater  than  the  new  value  of  Ar,  the  observation  correspond- 
ing to  the  largest  of  them  is  rejected.  This  process  is  repeated 
and  observations  rejected  one  at  a  time  until  a  value  of  Ar  is  ob- 
tained that  is  greater  than  any  of  the  residuals. 

When  more  than  one  residual  is  greater  than  the  computed 
value  of  Ay,  only  the  observation  corresponding  to  the  largest 
of  them  should  be  rejected  without  further  study.  The  rejection 
of  a  single  observation  from  the  given  series  changes  the  arith- 
metical mean,  and  hence  all  of  the  residuals  and  the  value  of  E 
computed  from  them.  If  r  and  r'  are  the  residuals  corresponding 
to  the  same  observation  before  and  after  the  rejection  of  a  more 
faulty  observation,  and  if  Ar  and  Ar'  are  the  corresponding 
limiting  errors,  it  may  happen  that  r'  is  less  than  A/,  although  r 
is  greater  than  Ay.  Hence  the  second  application  of  the  criterion 
may  show  that  a  given  observation  should  be  retained  notwith- 
standing the  fact  that  its  residual  was  greater  than  the  limiting 
error  in  the  first  application,  provided  an  observation  with  a 
larger  residual  was  excluded  on  the  first  trial. 

To  facilitate  the  computation  of  Ay,  the  values  of  T  corre- 
sponding to  a  number  of  different  values  of  N  have  been 
interpolated  from  Table  XII  and  entered  in  the  second  column 
of  Table  XIII. 

For  the  purpose  of  illustration,  suppose  that  ten  micrometer 
settings  have  been  made  on  the  same  mark  and  recorded,  to  the 
nearest  tenth  of  a  division  of  the  micrometer  head,  as  in  the  first 
column  of  the  following  table. 


130 


THE   THEORY  OF  MEASUREMENTS       [ART.  71 


Obs. 

r 

r' 

2.567 

+0.0118 

2.559 

+0.0038 

+0.0051 

2.556 

+0.0008 

+0.0021 

2.552 

-0.0032 

-0.0019 

2.551 

-0.0042 

-0.0029 

2.553 

-0.0022 

-0.0009 

2.555 

-0.0002 

+0.0011 

2.548 

-0.0072 

-0.0059 

2.554 

-0.0012 

+0.0001 

2.557 

+0.0018 

+0.0031 

x  =2.5552 

[r]  =  0.0364 

[r>]  =  0.0231 

z'=2.5539 

#  =  0.0032 

#'  =  0.0023 

IF  =  2.  91 

T'  =  2.84 

Ar  =  0.0093 

A/  =  0.0065 

The  residuals,  computed  from  the  mean  x,  are  given  under  r. 
The  probable  error  E}  computed  from  [r]  by  the  first  of  equations 
(34),  with  the  aid  of  Table  XV,  is  0.0032.  The  value  of  T  corre- 
sponding to  ten  observations  is  2.91  from  Table  XIII,  and  the 
limiting  error  Ay  is  equal  to  0.0093.  Since  this  is  less  than  the 
residual  0.0118,  the  corresponding  observation  (2.567)  should  be 
rejected  from  the  series. 

The  mean  of  the  retained  observations,  xi,  is  2.5539,  and  the 
corresponding  residuals  are  given  under  r'  in  the  third  column  of 
the  above  table.  The  new  value  of  the  limiting  error  (A/),  com- 
puted by  the  same  method  as  above,  is  0.0065.  Since  none  of 
the  new  residuals  are  larger  than  this,  the  nine  observations  left 
by  the  first  application  of  the  criterion  should  all  be  retained. 

71.  Precision  of  Direct  Measurements.  —  The  first  step  in 
the  reduction  of  a  series  of  direct  observations  is  the  correction 
of  all  known  systematic  errors  and  the  test  of  the  completeness  of 
this  process  by  the  criteria  of  article  sixty-eight.  In  general,  the 
systematic  errors  represent  small  variations  of  otherwise  constant 
errors;  and,  in  making  the  preliminary  corrections,  it  is  best  to 
consider  only  this  variable  part,  i.e.,  the  corrections  are  so  applied 
that  all  of  the  corrected  observations  are  left  with  exactly  the 
same  constant  errors.  Thus,  suppose  that  the  temperature  of  a 
scale  is  varying  slowly  during  a  series  of  observations,  and  is 
never  very  near  to  the  temperature  at  which  the  scale  is  standard. 
It  is  better  to  correct  each  observation  to  the  mean  temperature 
of  the  scale  and  leave  the  larger  correction,  from  mean  to  standard 


ART.  71]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  131 

temperature,  until  it  can  be  applied  to  the  arithmetical  mean  in 
connection  with  the  corrections  for  other  strictly  constant  errors. 
This  is  because  the  systematic  variations  in  the  length  of  the 
scale  are  so  small  that  the  unavoidable  errors  in  the  observed 
temperatures  and  the  adopted  coefficient  of  expansion  of  the  scale 
can  produce  no  appreciable  effect  on  the  corrections  to  mean 
temperature.  The  effect  of  these  errors  on  the  larger  correction 
from  mean  to  standard  temperature  is  more  simply  treated  in 
connection  with  the  arithmetical  mean  than  with  the  individual 
observations. 

Let  01,  02,  .  .  .  ,  on  represent  a  series  of  direct  observations 
corrected  for  all  known  systematic  errors  and  satisfying  the 
criteria  of  accidental  errors.  We  have  seen  that  the  most  prob- 
able value  that  we  can  assign  to  the  numeric  of  the  measured  mag- 
nitude, on  the  basis  of  such  a  series,  is  given  by  the  relation 

x  =  om  +  c'+c"+  •  -  •  +cfe>,  (102) 

where  om  is  the  arithmetical  mean  of  the  o's,  and  the  c's  represent 
corrections  for  strictly  constant  errors.  If  the  c's  could  be  deter- 
mined with  absolute  accuracy,  or  even  within  limiting  errors  that 
are  negligible  in  comparison  with  the  accidental  errors  of  the  o's, 
the  only  uncertainty  in  the  above  expression  for  x  would  be  that 
due  to  the  accidental  error  of  om.  Hence,  by  equations  (103),  if 
Ex  and  Em  are  the  probable  errors  of  x  and  om,  respectively,  we 
should  have 


*.  =  *_  =  0.674  Vy  '.'        (HO) 

•   .  »    . 

If  we  follow  the  usual  practice  and  regard  the  probable  error  of  a 
quantity  as  a  measure  of  the  accidental  errors  of  the  observations 
from  which  it  is  directly  computed,  equation  (110)  still  holds 
when  the  accidental  errors  of  the  c's  are  not  negligible;  but,  as  we 
shall  see,  Ex  is  no  longer  a  complete  measure  of  the  precision  of  x 
in  such  cases. 

In  practice  each  of  the  c's  must  be  computed,  on  theoretical 
grounds,  from  subsidiary  observations  with  the  aid  of  physical 
constants  that  have  been  previously  determined  by  direct  or 
indirect  measurements.  For  the  sake  of  brevity  the  quantities 
on  which  the  c's  depend  will  be  called  correction  factors.  Since  all 
of  them  are  subject  to  accidental  errors,  the  computed  c's  are 
affected  by  residual  errors  of  indeterminate  sign  and  magnitude. 


132  THE   THEORY   OF  MEASUREMENTS       [ART.  71 

When  the  probable  errors  of  the  correction  factors  are  known  the 
probable  errors  of  the  c's  may  be  computed  by  the  laws  of  propa- 
gation of  errors  with  the  aid  of  the  correction  formulae  by  which 
the  c's  are  determined. 

Equation  (102)  gives  x  as  a  continuous  sum  of  om  and  the  c's. 
Consequently,  if  we  represent  the  probable  errors  of  the  c's  by 
Eit  E2,  .  .  .  ,  Eq,  respectively,  we  have  by  equation  (76),  article 
fifty-eight, 

Rx2  =  Em*  +  Ei*  +  •  •  •  +Eq*,  (111) 

wnere  Rx  is  the  resultant  probable  error  of  x  due  to  the  correspond- 
ing errors  of  om  and  the  c's.  To  distinguish  Rx  from  the  probable 
error  EX)  which  depends  only  on  the  accidental  error  of  om,  we 
shall  call  it  the  precision  measure  of  x. 

Although  equation  (111)  is  simple  in  form,  the  separate  compu- 
tation of  the  E'SJ  from  the  errors  of  the  correction  factors  on  which 
they  depend,  is  frequently  a  tedious  process.  Moreover  several 
of  the  c's  may  depend  on  the  same  determining  quantities.  Con- 
sequently the  computation  of  x  and  Rx  is  frequently  facilitated  by 
bringing  the  correction  factors  into  the  equation  for  x  explicitly, 
rather  than  allowing  them  to  remain  implicit  in  the  c's.  Thus, 
if  a,  )8,  .  .  .  ,  p  represent  the  correction  factors  on  which  the  c's 
depend,  equation  (102)  may  be  put  in  the  form 

x  =  F(om,a,0,  .  .  .  ,  P).  (112) 

Hence,  by  equation  (81),  article  sixty, 


where  Ea,  Ep,  etc.,  are  the  probable  errors  of  a,  ft,  etc. 

For  example,  suppose  that  om  represents  the  mean  of  a  num- 
ber of  observations  of  the  distance  between  two  parallel  lines 
expressed  in  terms  of  the  divisions  of  the  scale  used  in  making 
the  measurements.  Let  t\  represent  the  mean  temperature  of  the 
scale  during  the  observations;  L  the  mean  length  of  the  scale 
divisions  at  the  standard  temperature  U,  in  terms  of  the  chosen 
unit;  a  the  coefficient  of  expansion  of  the  scale;  and  ft  the  angle 
between  the  scale  and  the  normal  to  the  lines.  Then,  if  the 
individual  observations  have  been  corrected  to  mean  temperature 
ti  before  computing  the  mean  observation  om,  the  best  approxima- 


ART.  71]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  133 

tion  that  we  can  make  to  the  true  distance  between  the  lines  is 
given  by  the  expression 

x  =  omL\l]+a(ti  -  t0)  I  —  £, 

in  which  the  correction  factors  L,  a,  /?,  fa,  and  to  appear  explicitly  , 
as  in  the  general  equation  (112).  A  more  detailed  discussion  of 
this  example  will  be  found  in  article  seventy-three. 

If  we  represent  the  separate  effects  of  the  errors  Em,  Ea,  .  .  .  , 
Ep  on  the  error  Rx  by  Dm,  Da,  D$,  .  .  .  ,  DPJ  respectively,  we 
have 

*•-£*•/  D-  -  SE*->  :.:i  '  D>  *  TPE»    <m> 

and  (113)  becomes 

R*2  =  Dm*  +  Da2  +  Df  +  -  -  -  +  DP2.  (115) 

In  some  cases  the  fractional  effects 

_Drn,  _D«.  .  _D, 

m~   x   '       a~   x  '  '      '  '     p~  x 

can  be  more  easily  computed  numerically  than  the  corresponding 
D's.  When  this  occurs,  the  fractional  precision  measure 


is  first  computed  and  then  Rx  is  determined  by  the  relation 

Rx  =  x-Px.  (117) 

While  equations  (112)  to  (117)  are  apparently  more  complicated 
than  (102)  and  (111),  they  generally  lead  to  more  simple  numerical 
computations.  Moreover  the  probable  errors  of  some  of  the 
correction  factors  are  frequently  so  small  that  they  produce  no 
appreciable  effect  on  Rx.  When  either  equation  (115)  or  (116)  is 
used,  such  cases  are  easily  recognized  because  the  corresponding 
D's  or  P's  are  negligible  in  comparison  with  Dm  or  Pm.  Obvi- 
ously the  same  condition  applies  to  the  E's  in  equation  (111),  but 
the  numerical  computation  of  either  the  D's  or  the  P's  is  generally 
more  simple  than  that  of  the  E's  in  (111)  because  approximate 
values  of  om  and  the  correction  factors  may  be  used  in  evaluat- 
ing the  differential  coefficients  in  (114).  The  allowable  degree  of 
approximation,  the  limit  of  negligibility  of  the  D's,  and  some  other 


134  THE  THEORY  OF  MEASUREMENTS       [ART.  71 

details  of  the  computation  will  be  discussed  more  extensively 
in  the  next  article. 

If  the  true  numeric  of  the  measured  magnitude  is  represented 
by  Xj  the  final  result  of  a  series  of  direct  measurements  may  be 
expressed  in  the  form 

X  =  x±Rx,  (118) 

where  x  is  the  most  probable  value  that  can  be  assigned  to  X  on 
the  basis  of  the  given  observations,  and  Rx  is  the  precision  measure 
of  x.  In  practice  x  may  be  computed  by  either  equation  (102) 
or  (112),  or  the  arithmetical  mean  of  the  individually  corrected 
observations  may  be  taken,  and  Rx  is  given  by  equations  (111), 
(115),  or  (117),  the  choice  of  methods  depending  on  the  nature 
of  the  given  data  and  the  preference  of  the  computer. 

The  exact  significance  of  equation  (118)  should  be  carefully 
borne  in  mind,  and  it  should  be  used  only  when  the  implied  condi- 
tions have  been  fulfilled.  Briefly  stated,  these  conditions  are  as 
follows : 

1st.  The  accidental  errors  of  the  observations  on  which  x 
depends  follow  the  general  law  of  such  errors. 

2nd.  A  careful  study  of  the  methods  and  apparatus  used  has 
been  made  for  the  purpose  of  detecting  all  sources  of  constant 
or  systematic  errors  and  applying  the  necessary  corrections. 

3rd.  The  given  value  of  x  is  the  most  probable  that  can  be 
computed  from  the  observations  after  all  constant  errors,  system- 
atic errors,  and  mistakes  have  been  as  completely  removed  as 
possible. 

4th.  The  resultant  effect  of  all  sources  of  error,  whether  acci- 
dental errors  of  observation  or  residual  errors  left  by  the  correc- 
tions for  constant  errors,  is  as  likely  to  be  less  than  Rx  as  greater 
than  Rx. 

The  expressions  in  the  form  X  =  x  ±  Ex,  used  in  preceding 
chapters,  are  not  violations  of  the  above  principles  because,  in 
those  cases,  we  were  discussing  only  the  effects  of  accidental 
errors  and  the  observations  were  assumed  to  be  free  from  all  con- 
stant errors  and  mistakes.  Such  ideal  conditions  never  occur  in 
practice.  Consequently  Rx  should  not  be  replaced  by  Ex  in 
expressing  the  result  of  actual  measurements  in  the  form  of  equa- 
tion (118),  unless  it  can  be  shown  by  equation  (115),  and  the  given 
data  that  the  sum  of  the  squares  of  the  D's  corresponding  to  all 
of  the  correction  factors  is  negligible  in  comparison  with  Z)m2. 


ART.  72]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  135 

In  the  latter  case  Ex  and  Rx  are  identical  as  may  be  easily  seen 
by  comparing  equations  (110),  (111),  and  (115). 

72.  Precision  of  Derived  Measurements.  —  When  a  desired 
numeric  Z0  is  connected  with  the  numerics  Xi,  X2,  .  .  .  ,  Xq 
of  a  number  of  directly  measured  magnitudes  by  the  relation 

XQ  =  F  (Xi,  X%,    .     .     .    ,    Xq), 

the  most  probable  value  that  we  can  assign  to  XQ  is  given  by  the 
expression 

x0  =  F(x1,xt,  .  .  .  ,  xq),  (119) 

where  the  x's  are  the  most  probable  values  of  the  X's  with  corre- 
sponding subscripts.  Each  of  the  component  x's,  together  with 
its  precision  measure,  can  be  computed  by  the  methods  of  the  pre- 
ceding article.  The  precision  measure  of  XQ  may  be  computed 
with  the  aid  of  equation  (81),  article  sixty,  by  replacing  the  E's  in 
that  equation  by  the  R's  with  corresponding  subscripts. 

Sometimes  the  numerical  computations  are  simplified  and  the 
discussion  is  clarified  by  bringing  the  direct  observations  and  the 
correction  factors  explicitly  into  the  expression  for  XQ.  If  oa, 
Ob,  .  .  .  ,  Op  are  the  arithmetical  means  of  the  direct  observa- 
tions, after  correction  for  systematic  errors,  on  which  Xi,  xz,  .  .  .  , 
xq  respectively  depend,  and  a,  /?,  .  .  .  ,  p  are  the  correction 
factors  involved  in  the  constant  errors  of  the  observations,  equa- 
tion (119)  may  be  put  in  the  form 

x0  =  d  (oa,  ob,  .  .  .  ,  op,  a,  j8,  .  .  .  ,  p).  (120) 

The  function  6  is  always  determinable  when  the  function  F  in 
(119)  is  given  and  the  correction  formulae  for  the  constant  errors 
are  known. 

Representing  the  precision  measure  of  XQ  by  R0,  and  adopting 
an  obvious  extension  of  the  notation  of  the  preceding  article,  we 
have,  by  equation  (81), 


Introducing  the  separate  effects  of  the  E's, 

*-£*•'  '  '  '  =  »*=l^' 

(121)  becomes 


*'  '  '  '  ;  »'-*-  (122) 


.    (123) 


136  THE  THEORY  OF  MEASUREMENTS       [ART.  72 

The  fractional  effects  of  the  E's  are 

P  _£«.        .  P  =5*.    P  =^.        .  P  _A? 

^°  "  XQ  '  '  p       x0  '        a       Z0  '  p  "  XQ  ' 

and  the  fractional  precision  measure  of  x0  is  given  by  the  relation 


XQ 

When  the  numerical  computation  of  the  P's  is  simpler  than  that 
of  the  D's,  PO  is  first  computed  by  equation  (124)  and  then  RQ 
is  determined  by  the  relation 

#o  =  z0Po.  (125) 

The  expression  of  the  final  result  of  the  observations  and  com- 
putations in  the  form 

XQ  =  XQ  ±  RQ 

has  exactly  the  same  significance  with  respect  to  XQ,  XQ,  and  RQ 
that  (118)  has  with  respect  to  X,  x,  and  Rx.  It  should  not  be 
used  until  all  of  the  underlying  conditions  have  been  fulfilled  as 
pointed  out  in  the  preceding  article.  Confusion  of  the  precision 
measure  R0  with  the  probable  error  E0)  and  insufficient  rigor  in 
eliminating  constant  errors  have  led  many  experimenters  to  an 
entirely  fictitious  idea  of  the  precision  of  their  measurements. 

When  the  correction  factors  are  explicitly  expressed  in  the 
reduction  formulae,  as  in  equations  (112)  and  (120),  the  only 
difference  between  the  expressions  for  direct  and  derived  measure- 
ments is  seen  to  lie  in  the  greater  number  of  directly  observed 
quantities,  oa,  o&,  etc.,  that  appear  in  the  latter  equation.  The 
same  methods  of  computation  are  available  in  both  cases  and  the 
following  remarks  apply  equally  well  to  either  of  them. 

For  practical  purposes,  the  precision  measure  R  is  computed 
to  only  two  significant  figures  and  the  corresponding  x  is  carried 
out  to  the  place  occupied  by  the  second  significant  figure  in  R. 
The  reasons  underlying  this  rule  have  been  fully  discussed  in 
article  forty-three,  in  connection  with  the  probable  error,  and 
need  not  be  repeated  here.  In  computing  the  numerical  value 
of  the  differential  coefficients  in  equations  (113),  (114),  (121),  and 
(122),  the  observed  components,  om,  oa,  o&,  etc.,  and  the  correc- 
tion factors,  a,  £,  etc.,  are  rounded  to  three  significant  figures, 
and  those  that  affect  the  result  by  less  than  one  per  cent  are  neg- 
lected. This  degree  of  approximation  will  always  give  R  within 


ART.  72]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  137 

one  unit  in  the  second  significant  figure  and  usually  decreases  the 
labor  of  computation. 

Generally  the  components  om,  oa,  ob,  etc.,  represent  the  arith- 
metical means  of  series  of  direct  observations  that  have  been 
corrected  for  systematic  errors.  In  such  cases  the  corresponding 
probable  errors  Emt  Ea,  Eb,  etc.,  can  be  computed,  by  equations 
in  the  form  of  (110),  from  the  residuals  determined  by  equations 
in  the  form  of  (103),  with  the  aid  of  the  observations  on  which 
the  o's  depend.  If  the  observations  are  sufficiently  numerous, 
the  computation  of  the  .27's.may  be  simplified  by  using  formulae 
depending  on  the  average  error  in  the  form 

E  =  0.845 fl=>  (34) 

n  Vn  —  1 

where  [f]  is  the  sum  of  the  residuals  without  regard  to  sign  and  n 
is  the  number  of  observations.  If  the  observations  on  which  any 
of  the  o's  depend  are  not  of  equal  weight,  the  general  mean  should 
be  used  in  place  of  the  arithmetical  mean  and  the  corresponding 
probable  errors  should  be  computed  by  equations  (41),  (42),  or 
(44),  depending  on  the  circumstances  of  the  observations. 

The  o's  in  equation  (120)  are  supposed  to  represent  simultane- 
ous values  of  the  directly  observed  magnitudes.  When  any  of 
these  quantities  are  continuous  functions  of  the  time,  or  of  any 
other  independent  variables,  it  frequently  happens  that  only  a 
single  observation  can  be  made  on  them  that  is  simultaneous 
with  the  other  components.  In  such  cases  this  single  observation 
must  be  used  in  place  of  the  corresponding  o  in  (120),  and  its 
probable  error  must  be  determined  for  use  in  equation  (122). 
For  the  latter  purpose,  it  is  sometimes  possible  to  make  an  auxil- 
iary series  of  observations  under  the  same  conditions  that  pre- 
vailed during  the  simultaneous  measurements  except  that  the 
independent  variables  are  controlled.  The  required  E  may  be 
assumed  to  be  equal  to  the  probable  error  of  a  single  observation 
in  the  auxiliary  series.  Consequently  it  may  be  computed  by 
formulae  in  the  form, 

E  =  0.674*  /W 
E  =  0.845 


n-  I 
or 

[r] 


138  THE  THEORY  OF  MEASUREMENTS       [ART.  72 

where  n  is  the  number  of  auxiliary  observations,  and  the  r's  are 
the  corresponding  residuals.  In  some  cases  this  simple  expedient 
is  not  available;  and  approximate  values  must  be  assigned  to  the 
E's  on  theoretical  grounds,  depending  on  the  nature  of  the  meas- 
urements; or  more  or  less  extensive  experimental  investigations 
must  be  undertaken  to  determine  their  values  more  precisely. 

Such  investigations  are  so  various  in  character  and  their  utility 
depends  so  much  on  the  skill  and  ingenuity  of  the  experimenter, 
that  a  detailed  general  discussion  of  them  would  be  impossible. 
They  may  be  illustrated  by  the  following  very  common  case. 
Suppose  that  one  of  the  components  in  equation  (120)  repre- 
sents the  gradually  changing  temperature  of  a  bath.  In  com- 
puting xQ  we  must  use  the  thermometer  reading  ot  taken  at  the 
time  the  other  components  are  observed.  The  errors  of  the  fixed 
points  of  the  thermometer  and  its  calibration  errors  enter  the 
equation  among  the  correction  factors  a,  /?,  etc.,  and  do  not  con- 
cern us  in  the  present  discussion.  In  order  to  determine  the 
probable  error  of  ot,  the  temperature  of  the  bath  may  be  caused 
to  rise  uniformly,  through  a  range  that  includes  ot,  by  passing  a 
constant  current  through  an  electric  heating  coil,  or  the  bath 
may  be  allowed  to  cool  off  gradually  by  radiation.  In  either  case 
the  rate  of  change  of  temperature  should  be  nearly  the  same  as 
prevailed  when  ot  was  observed.  A  series  of  corresponding  obser- 
vations of  the  time  T  and  the  temperature  t  are  made  under 
these  conditions,  and  the  empirical  relation  between  T  and  t  is 
determined  graphically  or  by  the  method  of  least  squares.  The 
probable  error  of  ot  may  be  assumed  to  be  equal  to  the  probable 
error  of  a  single  observation  of  t  in  this  series,  and  may  be  com- 
puted by  equation  (94),  article  sixty-four. 

Some  of  the  correction  factors  a,  ft,  etc.,  appearing  as  com- 
ponents in  equations  (112)  and  (120),  represent  subsidiary  obser- 
vations, and  some  of  them  represent  physical  constants.  The 
subsidiary  observations  may  be  treated  by  the  methods  outlined 
above.  When  the  highest  attainable  precision  is  desired,  the 
physical  constants,  together  with  their  probable  errors,  must  be 
determined  by  special  investigation.  In  less  exact  work  they 
may  be  taken  from  tables  of  physical  constants.  Such  tabular 
values  seldom  correspond  exactly  to  the  conditions  of  the  experi- 
ments in  hand  and  their  probable  errors  are  seldom  given. 
Generally  a  considerable  range  of  values  is  given,  and,  unless 


ART.  72]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  139 

there  is  definite  reason  in  the  experimental  conditions  for  the 
selection  of  a  particular  value,  the  mean  of  all  of  them  should  be 
adopted  and  its  probable  error  placed  equal  to  one-half  the  range 
of  the  tabular  values.  The  deviations  of  the  tabular  values  from 
the  mean  are  due  more  to  differences  in  experimental  conditions 
and  in  the  material  treated  than  to  accidental  errors.  Conse- 
quently a  probable  error  calculated  from  the  deviations  would 
have  no  significance  unless  these  differences  could  be  taken  into 
account.  The  selection  of  suitable  values  from  tables  of  physical 
constants  requires  judgment  and  experience,  and  the  general 
statements  above  should  not  be  blindly  followed.  In  many  cases 
the  original  sources  of  the  data  must  be  consulted  in  order  to 
determine  the  values  that  most  nearly  satisfy  the  conditions  of 
the  experiments  in  hand. 

In  good  practice  the  conditions  of  the  experiment  are  usually 
so  arranged  that  the  D's,  in  equation  (123),  corresponding  to  the 
direct  observations  oa,  o&,  etc.,  are  all  equal.  None  of  the  D's 
corresponding  to  correction  factors  should  be  greater  than  this 
limit,  but  it  sometimes  happens  that  some  of  them  are  much 
smaller.  Since  R0  is  to  be  computed  to  only  two  significant 
figures,  any  single  D  which  is  less  than  one-tenth  of  the  average 
of  the  other  D's  may  be  neglected  in  the  computation.  If  the 
sum  of  any  number  of  D's  is  less  than  one-tenth  of  the  average 
of  the  remaining  D's  they  may  all  be  neglected.  A  somewhat 
more  rigorous  limit  of  rejection  can  be  developed  for  use  in  plan- 
ning proposed  measurements,  but  it  is  scarcely  worth  while  in 
the  present  connection  since  the  correction  factors  and  all  other 
quantities  must  be  taken  as  they  occurred  in  the  actual  measure- 
ments, and  negligible  D's  are  very  easily  distinguished  by  inspec- 
tion after  a  little  experience. 

After  #o  has  been  determined,  x0  may  be  computed  by  either 
equation  (119)  or  (120).  If  (119)  is  used  the  x's  must  first  be 
determined  by  (102)  or  (112).  Sometimes  the  computation  may 
be  facilitated  by  using  a  modification  of  (120),  in  which  some  of 
the  correction  factors  appear  explicitly  while  others  are  allowed 
to  remain  implicit  in  the  z's  to  which  they  apply.  Such  cases 
cannot  be  treated  generally,  but  must  be  left  to  the  ingenuity  of 
the  computer.  Whatever  formula  is  used,  the  observed  quanti- 
ties and  the  correction  factors  should  be  expressed  by  sufficient 
significant  figures  to  give  the  computed  XQ  within  a  few  units  in 


140  THE  THEORY  OF  MEASUREMENTS       [ART.  73 

the  place  occupied  by  the  second  significant  figure  of  R0.  Occa- 
sionally the  total  effect  of  one  or  more  of  the  correction  factors  is 
less  than  this  limit  and  may  be  neglected  in  the  computation.  For 

f$  W  7? 

a  single  factor,  say  a,  this  is  the  case  when  —  a  is  less  than  ~ 

73.  Numerical  Example.  —  The  following  illustration  repre- 
sents a  series  of  measurements  taken  for  the  purpose  of  cali- 
brating the  interval  between  the  twenty-fifth  and  seventy-fifth 
graduations  on  a  steel  scale  supposed  to  be  divided  in  centimeters. 
The  observations  were  made  with  a  cathetometer  provided  with 
a  brass  scale  and  a  vernier  reading  to  one  one-thousandth  of  a 
division.  One  division  of  the  level  on  this  instrument  corre- 
sponds to  an  angular  deviation  of  3  X  10~4  radians,  and  the  ad- 
justments were  all  well  within  this  limit.  The  steel  scale  was 
placed  in  a  vertical  position  with  the  aid  of  a  plumb-line,  and, 
since  a  deviation  of  one-half,  millimeter  per  meter  could  have 
been  easily  detected,  the  error  of  this  adjustment  did  not  exceed 
5  X  10~4  radians.  Consequently  the  angle  between  the  two 
scales  was  not  greater  than  8  X  10~4  radians,  and  it  may  have 
been  much  smaller  than  this.  The  temperature  of  the  scales  was 
determined  by  mercury  in  glass  thermometers  hanging  in  loose 
contact  with  them.  The  probable  error  of  these  determinations 
was  estimated  at  five-tenths  of  a  degree  centigrade,  due  partly 
to  looseness  of  contact  and  partly  to  an  imperfect  knowledge  of 
the  calibration  errors  of  the  thermometers. 

Twenty  independent  observations,  when  tested  by  the  last 
two  criteria  of  article  sixty-eight,  showed  no  evidence  of  the  pres- 
ence of  systematic  errors  or  mistakes.  Consequently  the  mean 
om,  in  terms  of  cathetometer  scale  divisions,  and  its  probable 
error  Em  were  computed  before  the  removal  of  constant  errors. 
The  following  numerical  data  represents  the  results  of  the  obser- 
vations and  the  known  calibration  constants  of  the  cathetometer. 

Mean  temperature  of  the  steel  scale,  T 20°  ±  0°.5  C. 

Mean  temperature  of  the  brass  scale,  ti 21°.3  =t  0°.5  C. 

Mean  of  twenty  observations  on  the  measured 

interval  in  terms  of  brass  scale  divisions,  om. .  50.0051  db  0.0015  scale  div. 
Mean  length,  at  standard  temperature,  of  the 

brass  scale  divisions  in  the  interval  used,  S. .  0.999853  d=  0.000024  cm. 

Standard  temperature  of  brass  scale,  t0 15°.0  C. 

Coefficient  of  linear  expansion  of  brass  scale,  a.  (182  ±  12)  X  10~7. 

Angle  between  two  scales,  /3,  less  than 8  X  10-4  rad. 


ART.  731  DISCUSSION  OF  COMPLETED  OBSERVATIONS  141 

The  most  probable  value  that  can  be  assigned  to  the  measured 
interval  is  given  by  the  expression 


Since  ft  is  a  very  small  angle,  --  -  may  be  treated  by  the  approxi- 

COS  p 

mate  formulae  of  Table  VII,  and  the  above  expression  becomes 


where 

t  =  fa-to. 


The  quantity  S  (1  -f-  at)  is  very  nearly  equal  to  unity.  Hence, 
neglecting  small  quantities  of  the  second  and  higher  orders,  the 
correction  due  to  the  angle  ft  is 


<  0.000016. 

Since  this  is  less  than  two  per  cent  of  the  probable  error  of  om,  it  is 
negligible  in  comparison  with  the  accidental  errors  of  observation. 
Consequently  the  precision  of  x  is  not  increased  by  retaining  the 
term  involving  ft,  and  we  may  put 

x  =  OmS  (1  +  at).  (a) 

The  probable  error  of  tQ  is  zero,  because  the  accidental  errors  of 
the  temperature  observations,  made  during  the  calibration  of  the 
brass  scale,  are  included  in  the  probable  errors  of  S  and  a  com- 
puted by  the  method  of  article  sixty-five.  Consequently  the 
probable  error  of  t  is  equal  to  that  of  fa,  and  we  have 

t  =  6°,3  ±  0°.5  C. 

In  the  present  case  equation  (115)  is  the  most  convenient  for 
computing  the  precision  measure  ,.RX  of  x.  Only  two  significant 
figures  are  to  be  retained  in  the  separate  effects  computed  by 
equation  (114).  Consequently  the  factor  (1  +  at)  may  be  taken 
equal  to  unity,  and  the  numerical  values  of  om  and  S  may  be 
rounded  to  three  significant  figures  for  the  purpose  of  this  com- 
putation. Thus,  taking  om  equal  to  50.0,  S  equal  to  1.00,  and 
the  other  data  as  given  above,  we  have 


142  THE   THEORY  OF  MEASUREMENTS       [ART.  73 


Dm=      -Em  =  S(l+  at)  Em=  1  X  Em=  0.0015. 

oom 

D,=  ~QEt=om(l  +  at)  E,=  50  X  Ea  =  0.0012. 

do 

Da=~Ea  =  OmStEa  =  50  X  6.3  X  Ea  =  0.00038. 

da 


m  =50  X  182  X  10~7  X  Et  =  0.00046. 

ot 

Dm2=  225.0  X  10~8 
A,2  =  144.0  X  10~8 
Z>«2  =  14.4  X  10~8 
A2  =  21.2  X  10~8 
[D2]  =  404.6  X  10~8 
Hence,  by  equation  (115), 

Rx*=  [D2]  =  404.6  X  10-8, 

JBX  =  V404.6  X  10-8  =  0.0020. 

For  the  purpose  of  computing  x,  it  is  convenient  to  put  the 
given  data  in  the  form 

Om=  50  (1+0.000102), 
S  =  1-  0.000147, 
at  =  0.000115. 

Then,  by  equation  (a), 

x  =  50  (1  +  0.000102)  (1  -  0.000147)  (1  +  0.000115), 
and  by  formula  7,  Table  VII, 

x  =  50  (1  +  0.000102  -  0.000147  +  0.000115) 
=  50  (1  +  0.00007) 
=  50.0035. 

This  method  of  computation,  by  the  use  of  the  approximate 
formulae  of  Table  VII,  gives  x  within  less  than  one  unit  in  the  last 
place  held,  and  is  much  less  laborious  than  the  use  of  logarithms. 
Since  the  length  S  of  the  cathetometer  scale  divisions  is  given 
in  centimeters,  the  computed  values  of  x  and  Rx  are  also  expressed 
in  centimeters  and  our  uncertainty  regarding  the  true  distance  L 
between  the  twenty-fifth  and  the  seventy-fifth  graduations  of  the 
steel  scale  is  definitely  stated  by  the  expression 

L  =  50.0035  d=  0.0020  centimeters, 
at  the  temperature 

Tr  =  20°.0±0°.5C. 


ART.  73]  DISCUSSION  OF  COMPLETED  OBSERVATIONS  143 

The  above  discussion  shows  that  the  precision  of  the  result 
would  not  have  been  materially  increased  by  a  more  accurate 
determination  of  T,  fa,  and  a,  since  the  effects  of  the  errors  of 
these  quantities  are  small  in  comparison  with  that  of  the  errors 
of  om  and  S.  The  probable  error  of  om  might  have  been  reduced 
by  making  a  larger  number  of  observations  and  taking  care  to 
keep  the  instrument  in  adjustment  within  one-tenth  of  a  level 
division  or  less.  But  the  given  value  of  Em  is  of  the  same  order 
of  magnitude  as  the  least  count  of  the  vernier  used,  and,  since 
each  observation  represents  the  difference  of  two  scale  readings, 
it  would  not  be  decreased  in  proportion  to  the  increased  labor  of 
observation.  Moreover,  the  terms  Dm  and  D8  in  the  above  value 
of  Rx  are  nearly  equal  in  magnitude,  and  it  would  not  be  worth 
while  to  devote  time  and  labor  to  the  reduction  of  one  of  them 
unless  the  other  could  be  reduced  in  like  proportion. 


CHAPTER  XI. 
DISCUSSION   OF  PROPOSED   MEASUREMENTS. 

74.  Preliminary  Considerations.  —  The  measurement  of  a 
given  quantity  may  generally  be  carried  out  by  any  one  of  several 
different,  and  more  or  less  independent,  methods.  The  available 
instruments  usually  differ  in  type  and  in  functional  efficiency.  A 
choice  among  methods  and  instruments  should  be  determined  by 
the  desired  precision  of  the  result  and  the  time  and  labor  that  it  is 
worth  while  to  devote  to  the  observations  and  reductions. 

Since  the  labor  of  observation  and  the  cost  of  instruments  in- 
crease more  rapidly  than  the  inverse  square  of  the  precision 
measure  of  the  attained  result,  a  considerable  waste  of  time  and 
money  is  involved  in  any  measurement  that  is  executed  with 
greater  precision  than  is  demanded  by  the  use  to  which  the  result 
is  to  be  put.  On  the  other  hand,  if  the  precision  attained  is  not 
sufficient  for  the  purpose  in  hand,  the  measurement  must  be 
repeated  by  a  more  exact  method.  Consequently  the  labor  and 
expense  of  the  first  determination  contributes  very  little  to  the 
final  result  and  the  waste  is  quite  as  great  as  in  the  preceding 
case.  Sometimes  the  expense  of  a  second  determination  is 
avoided  by  using  the  inexact  result  of  the  first,  but  such  a  saving 
is  likely  to  prove  disastrous  unless  the  uncertainty  of  the  adapted 
data  is  duly  considered. 

In  general  the  greatest  economy  is  attained  by  so  planning 
and  executing  the  measurement  that  the  result  is  given  with  the 
desired  precision  and  neglecting  all  refinements  of  method  and 
apparatus  that  are  not  essential  to  this  end.  While  these  con- 
siderations have  greater  weight  in  connection  with  measurements 
carried  out  for  practical  purposes  they  should  never  be  neglected 
in  planning  investigations  undertaken  primarily  for  the  advance- 
ment of  science.  In  the  former  case  the  cost  of  necessary  measure- 
ments may  represent  an  appreciable  fraction  of  the  expense  of 
a  proposed  engineering  enterprise  and  must  be  taken  into  account 
in  preparing  estimates.  In  the  latter  case  there  is  no  excuse  for 
burdening  the  limited  funds  available  for  research  with  the  expense 

144 


ART.  75]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  145 

of  ill-contrived  and  haphazard  measurements.  The  precision 
requirements  may  be,  and  indeed  usually  are,  quite  different  in 
the  two  cases,  'but  the  same  process  of  arriving  at  suitable  methods 
applies  to  both. 

75.  The  General  Problem.  —  In  its  most  general  form  the 
problem  may  be  stated  as  follows :  Required  the  magnitude  of  a 
quantity  X  within  the  limits  ±  R,  X  being  a  function  of  several 
directly  measured  quantities  X\,  X2,  etc. ;  within  what  limits  must 
we  determine  the  value  of  each  of  the  components  X\,  Xz,  etc.? 
In  discussing  this  problem,  all  sources  of  error  both  constant  and 
accidental  must  be  taken  into  account.  For  this  purpose  the 
various  methods  available  for  the  measurement  of  the  several 
components  are  considered  with  regard  to  the  labor  of  execution 
and  the  magnitude  of  the  errors  involved  as  well  as  with  regard  to 
the  facility  and  accuracy  with  which  constant  errors  can  be  removed. 

After  such  a  study,  certain  definite  methods  are  adopted  pro- 
visionally, and  examined  to  determine  whether  or  not  the  re- 
quired precision  in  the  final  result  can  be  attained  by  their  use. 
As  the  first  step  in  this  process,  the  function  that  gives  the  rela- 
tion between  X  and  the  components,  Xi,  X2,  etc.,  is  written  out 
in  its  most  complete  form  with  all  correction  factors  explicitly 
represented.  Thus,  as  in  article  seventy-two,  the  most  probable 
value  of  the  quantity  X  may  be  expressed  in  the  form 

XQ  =  0(oa,obj  .  .  .  ,  0p,a,/3,  .  .  .  ,  p),  (120) 

where  the  o's  represent  observed  values  of  X\t  X2,  etc.,  and  a,  /3, 
.  .  .  ,  p,  represent  the  factors  on  which  the  corrections  for  con- 
stant errors  depend  as  pointed  out  in  connection  with  equation 
(112),  article  seventy-one. 

The  form  of  the  function  0,  and  the  nature  and  magnitude  of 
the  correction  factors  appearing  in  it,  will  depend  on  the  nature 
of  the  proposed  methods  of  measurement.  Since  all  detectable 
constant  errors  are  explicitly  represented  by  suitable  correction 
factors,  all  of  the  quantities  appearing  in  the  function  0  may  be 
treated  as  directly  measured  components  subject  to  accidental 
errors  only.  Hence  the  problem  reduces  to  the  determination 
of  the  probable  errors  within  which  each  of  the  components  must 
be  determined  in  order  that  the  computed  value  of  XQ  may  come 
out  with  a  precision  measure  equal  to  the  given  magnitude  RQ. 
If  all  of  the  components  can  be  determined  within  the  limits  set 


146  THE   THEORY  OF  MEASUREMENTS       [ART.  76 

by  the  probable  errors  thus  found,  without  exceeding  the  limits 
of  time  and  expense  imposed  by  the  preliminary  considerations, 
the  provisionally  adopted  methods  are  adequate  for  the  purpose 
in  hand  and  the  measurements  may  be  carried  out  with  con- 
fidence that  the  final  result  will  be  precise  within  the  required 
limits.  When  one  or  more  of  the  components  cannot  be  deter- 
mined within  the  limits  thus  set  without  undue  labor  or  expense, 
the  proposed  methods  must  be  modified  in  such  a  manner  that  the 
necessary  measurements  will  be  feasible. 

76.  The  Primary  Condition.  —  The  present  problem  is,  to 
some  extent,  the  inverse  of  that  treated  in  articles  seventy-one 
and  seventy-two.  In  the  latter  case  the  given  data  represented 
the  results  of  completed  series  of  observations  on  the  several 
component  quantities  appearing  in  the  function  0,  together  with 
their  respective  probable  errors.  The  purpose  of  the  analysis  was 
the  determination  of  the  most  probable  value  XQ  that  could  be 
assigned  to  the  measured  magnitude  and  the  precision  measure 
of  the  result.  In  the  present  case  approximate  values  of  x0  and 
the  components  in  6  are  given,  and  the  object  of  the  analysis  is 
the  determination  of  the  probable  errors  within  which  each  of  the 
components  must  be  measured  in  order  that  the  value  of  XQ, 
computed  from  the  completed  observations,  may  come  out  with  a 
precision  measure  equal  to  a  given  magnitude  R0. 

If  D0,  Db,  .  .  .  ,  Dp,  Da)  Dp,  .  .  .  ,  Dp  represent  the  separate 
effects  of  the  probable  errors  Ea,  Eb,  .  .  .  ,  Ep,  Ea,  Ep,  .  .  .  , 
Ep  of  the  components  oaj  ob,  .  .  .  ,  op,  a,  /3,  .  .  .  ,  p,  respec- 
tively, we  have,  as  in  article  seventy-two, 


and  the  primary  condition  imposed  on  these  quantities  is  given  by 
the  relation 
#o2  =  Da2  +  ZV  +  •  •  -  +  ZV  +  ZV  +  iy  +  -  -  .  +DP2.   (123) 

The  precision  measure  R0  and  approximate  values  of  the  com- 
ponents are  given  by  the  conditions  of  the  problem  and  the  pro- 
posed methods  of  measurement.  The  E's,  and  hence  also  the 
D's,  are  the  unknown  quantities  to  be  determined.  Conse- 
quently there  are  as  many  unknowns  in  equation  (123)  as  there 
are  different  components  in  the  function  0.  Obviously  the  problem 
is  indeterminate  unless  some  further  conditions  can  be  imposed 


ART.  77]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  147 

on  the  D's;  for  otherwise  it  would  be  possible  to  assign  an  infinite 
number  of  different  values  to  each  of  the  D's  which,  by  proper 
selection  and  combination,  could  be  made  to  satisfy  the  primary 
condition  (123). 

77.  The  Principle  of  Equal  Effects.  —  An  ideal  condition  to 
impose  on  the  D's  would  specify  that  they  should  be  so  determined 
that  the  required  precision  in  the  final  result  XQ  would  be  attained 
with  the  least  possible  expense  for  labor  and  apparatus.  Un- 
fortunately this  condition  cannot  be  put  into  exact  mathematical 
form  since  there  is  no  exact  general  relation  between  the  difficulty 
and  the  precision  of  measurements.  However,  it  is  easy  to  see 
that  the  condition  is  approximately  fulfilled  when  the  measure- 
ments are  so  made  that  the  D's  are  all  equal  to  the  same  magnitude. 
For,  the  probable  error  of  any  component  is  inversely  proportional 
to  the  square  root  of  the  number  of  observations  on  which  it 
depends  and  the  expense  of  a  measurement  increases  directly 
with  the  number  of  observations.  Consequently  the  expense 

Wa  of  the  component  oa  is  approximately  proportional  to  7^-5  or, 

•&a 
n/j  1 

since  r—  is  constant,  to  -^—9  .    Similar  relations  hold  for  the  other 
doa  Da2 

components.  Hence,  as  a  first  approximation,  we  may  assume 
that 

A2  A2  A2  A2 


where  W  is  the  total  expense  of  the  determination  of  x0,  and  A  is 
a  constant.  By  the  usual  method  of  finding  the  minimum  value 
of  a  function  of  conditioned  quantities,  the  least  value  of  W  con- 
sistent with  equation  (123)  occurs  when  the  D's  satisfy  (123)  and 
also  fulfill  the  relations 


_ 

dDa  "*    ^  dDa  = 

ML  +  *»**?-  o 

dDb  ^     *   dDb  - 


= 

SD   *    ^  dD 


148  THE  THEORY  OF  MEASUREMENTS       [ART.  77 

where  K  is  a  constant.     Introducing  the  expressions  for  R<?  and 
W  in  terms  of  the  D's,  differentiating,  and  reducing,  we  have 


and  by  equation  (123) 


where  AT  is  the  number  of  D's  in  (123)  or  the  equal  number  of 
components  in  the  function  6.  Consequently  equation  (123)  is 
fulfilled  and  the  condition  of  minimum  expense  is  approximately 
satisfied  when  the  components  are  so  determined  that  the  separate 
effects  of  their  probable  errors  satisfy  the  relation 

Da  =  Db  =  -  .  -  =  Da  =  Dp  =  •  •  •  =  -.          (127) 


Equation  (127)  is  the  mathematical  expression  of  the  principle 
of  equal  effects.  It  does  not  always  express  an  exact  solution  of 
the  problem,  since  A  is  seldom  strictly  constant;  but  it  is  the 
best  approximation  that  we  can  adopt  for  the  preliminary  com- 
putation of  the  D's  and  E's.  The  results  thus  obtained  will 
usually  require  some  adjustment  among  themselves  before  they 
will  satisfy  both  the  preliminary  considerations  and  the  primary 
condition  (123).  We  shall  see  that  the  necessary  adjustment  is 
never  very  great;  and,  in  fact,  that  a  marked  departure  from  the 
condition  of  equal  effects  is  never  possible  when  equation  (123)  is 
satisfied. 

Combining  equations  (122)  and  (127),  we  find 

E  —   ^°   .   ^  -  •     E  —   ^°   .  ^  • 

0    VAT  " de '     a    VN  ' de ' 


da 


w      Ro     i  . 

«*  =  ~~7=  '  ~^7T  > 


VN  <&'         VN  y, 

dob  5/3 


(128) 


Hence,  if  the  final  measurements  are  so  executed  that  the  probable 
errors  of  the  several  components  are  equal  to  the  corresponding 
values  given  by  equations  (128),  the  final  result  XQ,  computed  by 
equation  (120),  will  come  out  with  a  precision  measure  equal  to 


ART.  78]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  149 

the  specified  RQ,  and  the  condition  of  equal  effects  (127)  will  be 
fulfilled. 

In  computing  the  E's  by  equation  (128),  RQ  is  taken  equal  to 
the  given  precision  measure  of  XQ  and  N  is  placed  equal  to  the 

•J/3 

number  of  components  in  the  function  0.     The  derivatives  T— 

doa 

etc.,  are  evaluated  with  the  aid  of  approximate  values  of  the 
components  obtained  by  a  preliminary  trial  of  the  proposed 
methods  or  by  computation,  on  theoretical  grounds,  from  an 
approximate  value  of  XQ  and  a  knowledge  of  the  conditions  under 
which  the  measurements  are  to  be  made.  Since  only  two  sig- 
nificant figures  are  required  in  any  of  the  E's,  the  adopted  values 
of  the  components  may  be  in  error  by  several  per  cent,  without 
affecting  the  significance  of  the  results.  Moreover,  any  number 
of  components,  whose  combined  effect  on  any  derivative  is  less 
than  five  per  cent,  may  be  entirely  neglected  in  computing  that 
derivative.  Consequently  the  function  0  frequently  may  be  sim- 
plified very  much  for  the  purpose  of  computing  the  derivatives  and 
this  simplification  may  take  different  forms  in  the  case  of  differ- 
ent derivatives.  No  more  than  three  significant  figures  should  be 
retained  at  any  step  of  the  process  and  sometimes  the  required  pre- 
cision can  be  attained  with  the  approximate  formulae  of  Table  VII. 

Since  equation  (127)  is  an  approximation,  the  E's  derived  from 
equations  (128)  are  to  be  regarded  as  provisional  limits  for  the 
corresponding  components.  If  all  of  them  are  attainable,  i.e.,  if 
all  of  the  components  can  be  determined  within  the  provisional 
limits,  without  exceeding  the  limit  of  expense  set  by  the  prelim- 
inary considerations,  the  solution  of  the  problem  is  complete  and 
the  proposed  methods  are  suitable  for  the  work  in  hand. 

78.  Adjusted  Effects.  —  Generally  some  of  the  E's  given  by 
(128)  will  be  unattainable  in  practice  while  others  will  be  larger 
than  a  limit  that  can  be  easily  reached.  In  other  words,  it  will 
be  found  that  the  labor  involved  in  determining  some  of  the 
components  within  the  provisional  limit  is  prohibitive  while 
other  components  can  be  determined  with  more  than  the  pro- 
visional precision  without  undue  labor.  In  such  a  case  the  pro- 
visional limits  are  modified  by  increasing  the  E's  corresponding 
to  the  more  difficult  determinations  and  decreasing  the  E's  that 
correspond  to  the  more  easily  determinable  components  in  such  a 
way  that  the  combined  effects  satisfy  the  condition  (123). 


150  THE  THEORY  OF  MEASUREMENTS       [ART.  78 

The  maximum  allowable  increase  in  a  single  E  is  by  the  factor 
.     For,  taking  Ea  for  illustration, 


B0a 

and  consequently 


Hence  (123)  cannot  be  satisfied  unless  all  of  the  rest  of  the  D's 
are  negligibly  small.  For  example,  if  there  are  nine  components, 
VN  is  equal  to  three.  Consequently  no  one  of  the  E's  can  be 
increased  to  more  than  three  times  the  value  given  by  the  condi- 
tion of  equal  effects  if  (123)  is  to  be  satisfied.  When,  as  is  fre- 
quently the  case,  the  number  of  components  is  less  than  nine,  or 
when  more  than  one  of  the  E's  is  to  be  increased,  the  limit  of 
allowable  adjustment  is  much  less  than  the  above.  The  extent 
to  which  any  number  of  E's  may  be  increased  is  also  limited 
by  the  difficulty,  or  impossibility,  of  reducing  the  effects  of  the 
remaining  E's  to  the  negligible  limit. 

If  the  probable  errors  given  by  equations  (128)  can  be  modified, 
to  such  an  extent  that  the  corresponding  measurements  become 
feasible,  without  violating  the  condition  (123),  the  proposed 
methods  are  suitable  for  the  final  determination  of  XQ.  Other- 
wise they  must  be  so  modified  that  they  satisfy  the  conditions  of 
the  problem  or  different  methods  may  be  adopted  provisionally 
and  tested  for  availability  as  above. 

Sometimes  it  will  be  found  that  the  proposed  methods  are 
capable  of  greater  precision  than  is  demanded  by  equations  (128). 
In  such  cases  the  expense  of  the  measurements  may  be  reduced 
without  exceeding  the  given  precision  measure  of  XQ  by  using  less 
precise  methods.  But  such  methods  should  never  be  finally 
adopted  until  their  feasibility  has  been  tested  by  the  process  out- 
lined above. 

A  discussion  on  the  foregoing  lines  not  only  determines  the 
practicability  of  the  proposed  methods,  but  also  serves  as  a  guide 
in  determining  the  relative  care  with  which  the  various  parts  of 
the  work  should  be  carried  out.  For,  if  the  final  result  is  to  come 
out  with  a  precision  measure  RQ,  it  is  obvious  that  all  adjustments 
and  measurements  must  be  so  executed  that  each  of  the  com- 


ART.  79]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  151 

ponents  is  determined  within  the  limits  set  by  equations  (128), 
or  by  the  adjusted  E's  that  satisfy  (123). 

79.  Negligible  Effects.  —  In  the  preceding  article  it  was 
pointed  out  that  the  availableness  of  proposed  methods  of  meas- 
urement frequently  depends  on  the  possibility  of  so  adjusting  the 
E's  given  by  equations  (128)  that  they  are  all  attainable  and 
at  the  same  time  satisfy  the  primary  condition  (123).  Generally 
this  cannot  be  accomplished  unless  some  of  the  E's  can  be  reduced 
in  magnitude  to  such  an  extent  that  their  effect  on  the  precision 
measure  R0  is  negligible. 

On  account  of  the  meaning  of  the  precision  measure,  and  the 
fact  that  it  is  expressed  by  only  two  significant  figures,  it  is  obvi- 
ous that  any  D  is  negligible  when  its  contribution  to  the  value  of 

73 

#0  is  less  than  y^.     Thus,  if  Ri  is  the  value  of  the  right-hand 

member  of  equation  (123),  when  Da  is  omitted,  Da  is  negligible 
provided  „ 


or 

0. 
Squaring  gives 

0.81  Bo2  <  #i2, 
and  by  definition 

R<?  -  RS  =  D*. 
Consequently 

0.81  #o2  <  #o2  -  D*, 
and 

Z>a2<0.19#02, 
or 

Da  <  0.436  #o. 

Hence,  if  Da  is  less  than  0.436  #0,  it  will  contribute  lees  than  ten 
per  cent  of  the  value  of  RQ.  Since  the  true  error  of  x0  is  as  likely 
to  be  greater  than  R0  as  it  is  to  be  less  than  RQ,  a  change  of  ten 
per  cent  in  the  value  of  RQ  can  have  no  practical  importance. 
Consequently  Da  is  negligible  when  it  satisfies  the  above  condi- 
tion. However,  the  constant  0.436  is  somewhat  awkward  to 
handle,  and  if  Da  is  very  nearly  equal  to  the  limit  0.436  RQ,  the 
propriety  of  omitting  it  is  doubtful.  These  difficulties  may  be 
avoided  by  adopting  the  smaller  and  more  easily  calculated  limit 
of  rejection  given  by  the  condition 

D  =    RQ.  (129) 


152  THE   THEORY  OF  MEASUREMENTS       [ART.  79 

This  limit  corresponds  to  a  change  of  about  six  per  cent  in  the 
value  of  Ro  given  by  equation  (123),  and  is  obviously  safe  for  all 
practical  purposes.  Since  the  above  reasoning  is  independent  of 
the  particular  D  chosen,  the  condition  (129)  is  perfectly  general 
and  applies  to  any  one  of  the  D's  in  equation  (123). 

When  two  or  more  of  the  D's  satisfy  (129)  independently,  any 
one  of  them  may  be  neglected,  but  all  of  them  cannot  be  neg- 
lected without  further  investigation  for  otherwise  the  change  in 
Ro  might  exceed  ten  per  cent.  This  would  always  happen  if  all 

T~) 

of  the  D's  considered  were  very  nearly  equal  to  the  limit  ~^- 

o 

However,  by  analogy  with  the  above  argument,  it  is  obvious  that 
any  q  of  the  D's  are  simultaneously  negligible  when 


+  D22  +  .  .  .  +  D32  ==  Jflo,  (130) 

where  the  numerical  subscripts  1,  2,  .  .  .  ,  q  are  used  in  place 
of  the  literal  subscripts  occurring  in  equation  (123)  in  order  to 
render  the  condition  (130)  entirely  general.  Thus  DI  may  corre- 
spond to  any  one  of  the  D's  in  (123),  D2  to  any  other  one,  etc. 
By  applying  the  principle  of  equal  effects,  the  condition  (130) 
may  be  reduced  to  the  simple  form 

D,  =  D2=   ...   =  Dq  =  -  ^  (131) 

3  Vg 

If  some  of  the  D's  in  (131)  can  be  easily  reduced  below  the  limit 

•p 

— j=. ,  the  others  may  exceed  that  limit  somewhat  without  violating 
3  V  q 

the  condition  (130).  However,  equation  (131)  generally  gives  the 
best  practical  limit  for  the  simultaneous  rejection  of  a  number  of 
D's,  and  all  departures  from  it  should  be  carefully  checked  by  (130). 
To  illustrate  the  practical  application  of  the  foregoing  discussion, 
suppose  that  the  practicability  of  certain  proposed  methods  of 
measurement  is  to  be  tested  by  the  principle  of  equal  effects 
developed  in  article  seventy-seven.  Let  there  be  N  components 
in  the  function  0,  and  suppose  that  q  of  them,  represented  by 
ai,  «2,  .  .  .  ,  aq,  can  be  easily  determined  with  greater  precision 
than  is  demanded  by  equations  (128),  while  the  measurement 
of  the  remaining  N  —  q  components  within  the^limits  thus  set 
would  be  very  difficult.  Obviously  some  adjustment  of  the  E's 
given  by  (128)  is  desirable  in  order  that  the  labor  involved  in  the 
various  parts  of  the  measurement  may  be  more  evenly  balanced. 


ART.  79]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  153 


The  greatest  possible  increase  in  the  E's  corresponding  to  the 
N  —  q  difficult  components  will  be  allowable  when  the  E's  of  the 
q  easy  components  can  be  reduced  to  the  negligible  limit.  To 
determine  the  necessary  limits,  R0  is  taken  equal  to  the  given 
precision  measure  of  XQ,  and  the  negligible  D's  corresponding  to 
the  q  easy  components  are  determined  by  equation  (131).  Then 
by  equations  (122),  the  corresponding  E's  will  be  negligible  when 


E!=Z  -^ 

3  Vq 

1       1 
If 

dai 

E2  =  ±-^L< 

1 

w 

(132) 


A™  r 


J_^ 

6^ 
daq 


If  these  limits  can  be  attained  with  as  little  difficulty  as  the  pre- 
viously determined  E's  of  the  N  —  q  remaining  components,  the 
corresponding  D's  may  be  omitted  from  equation  (123)  during 
the  further  discussion  of  precision  limits. 

Since  q  of  the  D's  have  disappeared,  the  others  may  be  some- 
what increased  and  still  satisfy  the  primary  condition  (123). 
The  corresponding  new  limits  for  the  E's  of  the  difficult  components 
may  be  obtained  from  equations  (127)  and  (128)  by  replacing 
N  by  N  —  q.  If  these  new  limits  together  with  the  negligible 
limits  given  by  equations  (132)  can  all  be  attained,  without 
exceeding  the  expense  set  by  the  preliminary  considerations,  the 
proposed  methods  may  be  considered  suitable  for  the  final  deter- 
mination of  XQ  with  the  desired  precision.  Otherwise  new  methods 
must  be  devised  and  investigated  as  above. 

Equations  (132)  may  also  be  used  to  determine  the  extent  to 
which  mathematical  constants  should  be  carried  out  during  the 
computations.  For  this  purpose  the  components  «i,  0%  •  •  •  ,  ««, 
or  part  of  them,  represent  the  mathematical  constants  appearing 
in  the  function  8.  The  corresponding  E's,  determined  by  equa- 
tions (132),  give  the  allowable  limits  of  rejection  in  rounding  the 
numerical  values  of  the  constants  for  the  purpose  of  simplifying 


154 


THE   THEORY  OF  MEASUREMENTS       [ART.  79 


the  computations.  Thus,  suppose  that  the  volume  of  a  right 
circular  cylinder  of  length  L  and  radius  a  is  to  be  computed 
within  one-tenth  of  one  per  cent,  how  many  figures  should  be 
retained  in  the  constant  TT?  In  this  case 

n   /  \  17  9  T 

0  (Oa ,  •  •  •  ,  «,  •   •  •  )  =  y  =  *&lt, 
RQ  =  0.001  V  =  0.001 7ra2L, 
60      6V 


=  0.00105. 


0.001  7T 


If  TT  is  taken  equal  to  3.142  the  error  due  to  rounding  is  0.00041  — . 
Since  this  is  less  than  the  negligible  limit  Er,  four  significant 
figures  in  TT  are  sufficient  for  the  purpose  in  hand. 

It  sometimes  happens  that  the  total  effect  of  one  or  more  of  the 
components  in  the  function  0,  on  the  computed  value  of  x0,  is 
negligible  in  comparison  with  RQ.  This  will  obviously  be  the  case 
when 

60  RQ 

a^a  ^  IF' 


for  a  single  component  a  or  when 


KM  \2-L-/de 
z~~ai)  +  (^~~ 
dai  I  \da2 


da 


for  q  components.  Thus,  on  the  principle  of  equal  effects,  the 
components  «i,  <*2,  •  •  •  ,  <*3  will  be  simultaneously  negligible 
when  they  satisfy  the  conditions 

1      RQ        1 


*155i 


(133) 


RQ         1 

daz 

7">  1 

\7^'~d0~ 


Such  cases  frequently  arise  in  connection  with  the  components 
that  represent  correction  factors. 


ART.  80]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  155 

80.  Treatment  of  Special  Functions.  —  During  the  foregoing 
argument,  it  has  been  assumed  that  the  function  6  in  equation  (120) 
is  expressed  in  the  most  general  form  consistent  with  the  pro- 
posed methods  of  measurement.  Such  an  expression  involves  the 
explicit  representation  of  all  directly  measured  quantities,  and 
all  possible  correction  factors.  Part  of  the  latter  class  of  com- 
ponents represent  departures  of  the  proposed  methods  from  the 
theoretical  conditions  underlying  them,  and  others  depend  upon 
inaccuracies  in  the  adjustment  of  instruments.  In  practice  it 
frequently  happens  that  the  general  function  0  is  very  compli- 
cated, and  consequently  that  the  direct  discussion  of  precision 
as  above  is  a  very  tedious  process.  Under  these  conditions  it  is 
desirable  to  modify  the  form  of  the  function  in  such  a  manner  as 
to  facilitate  the  discussion. 

Sometimes  the  general  function  9  can  be  broken  up  into  a  series 
of  independent  functions  or  expressed  as  a  continuous  product 
of  such  functions.     Thus,  it  may  be  possible  to  express  6  in  the 
form 
XQ  =  6  (oa,  ob,  .  .  .,  a,  |8,  .  .  .) 

=  /i(ai,a2,  .  .  .  )±/2(&i,&2,  .  .  .  )±/3(ci,c2,  .  .  . 


or  in  the  form 

XQ  =  d  (Oa,  Ob)     . 


(134) 


(135) 


=  /i(ai,a2,  .  .  .  )  X/2(&i,&2,  .  •  •  )  X/3(ci,c2,  .  .  . 

X  ...  X  /„  (mi,  m2)  .  .  .  ), 
where  the  a's,  &'s,  .  .  .  ,  and  m's  represent  the  same  components, 
oa,  ob,  .  .  .  ,  a,  0,  .  .  .  ,  that  appear  in  6  by  a  new  and  more 
general  notation.  The  functions  /i,  /2,  .  .  .  ,  fn  may  take  any 
form  consistent  with  the  problem  in  hand,  but  the  precision  dis- 
cussion will  not  be  much  facilitated  unless  they  are  independent 
in  the  sense  that  no  two  of  them  contain  the  same  or  mutually 
dependent  variables.  Sometimes  the  latter  condition  is  imprac- 
ticable and  it  becomes  necessary  to  include  the  same  component 
in  two  or  more  of  the  functions.  Under  such  conditions  the  expan- 
sion has  no  advantage  over  the  general  expression  for  0,  unless 
the  effect  of  the  errors  of  each  of  the  common  components  can 
be  rendered  negligible  in  all  but  one  of  the  functions.  It  is 
scarcely  necessary  to  point  out  that  equations  (134)  and  (135) 
represent  different  problems,  and  that  if  it  were  possible  to  expand 


156  THE   THEORY  OF  MEASUREMENTS       [ART.  80 

the  same  function  0  in  both  ways,  the  component  functions  /i, 
/2,  •  •  •  ,  fn  would  be  different  in  the  two  cases. 
For  the  sake  of  convenience  let 

/I  (Oi,  «2,    •    •    •    )  =  2 
/2  (6l,   62,     .     .     .    )    =  ^2 


jfn  (Wi,m2,.     .     .    )    =  2 

Then  equation  (134)  may  be  written  in  the  form 

X0  =  Zi  ±  02  ±  2!3  ±    .    .    .    d=  2«,  (137) 

and  (135)  may  be  put  in  the  form 

x0  =  zlXzzXz3X  .  .  .  Xzn.  (138) 

First  consider  the  case  in  which  the  function  representing  the 
proposed  methods  of  measurement  has  been  put  in  the  form  of 
(137).  Since  the  precision  measure  follows  the  same  laws  of 
propagation  as  the  probable  error,  the  discussion  given  in  article 
fifty-eight  leads  to  the  relation 

#02  =  7^2  +  #22  +  Rf  +  _  m  +  Rn2}  (139) 

where  RQ  is  the  precision  measure  of  x0,  and  each  of  the  other  R's 
represents  the  precision  measure  of  the  z  with  corresponding  sub- 
script. Hence,  by  the  principle  of  equal  effects,  provisional 
values  of  the  R's  may  be  obtained  from  the  relation 

R,  =  R2  =  R,  =    .   .   .    =  Rn  =  A  .  (140) 


The  R's  having  been  determined  by  (140),  the  corresponding 
probable  errors  of  the  a's,  6's,  etc.,  may  be  computed  by  the 
methods  of  the  preceding  articles  with  the  aid  of  equations  (136). 
If  the  provisional  limits  of  precision  thus  found  are  not  all  attain- 
able with  approximately  equal  facility,  the  conditions  of  the 
problem  may  be  better  satisfied  by  moderately  adjusted  relative 
values  of  the  probable  errors  as  pointed  out  in  article  seventy- 
eight.  Obviously  the  adjusted  values  must  satisfy  equation  (139) 
if  the  value  of  x0  computed  by  (137)  is  to  come  out  with  a  pre- 
cision measure  equal  to  the  given  R0. 

When  the  function  representing  the  proposed  methods  can  be 
put  in  the  form  of  (138)  the  computation  is  facilitated  by  intro- 
ducing the  fractional  errors 

P0  =  «»;     Pl  =  «!;     P2  =  f2;...;     Pn  =  f"  •      (141) 

XQ  Zi  Zz  Zn 


ART.  81]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  157 

For,  by  the  argument  underlying  equation  (83),  article  sixty-two, 
Po2  =  Pi2  +  P22  +  Pa2  +  .  .  .  +  P«2,  (142) 

and,  by  the  principle  of  equal  effects,  provisional  values  of  the 
P's  are  given  by  the  relation 

Pi  =  P2  =  P3  =   .  .  .   =  P«  =  •£*=.  (143) 

Vn 

Since  RQ  and  approximate  values  of  the  components  are  given, 
PO  can  be  computed  with  sufficient  accuracy  with  the  aid  of 
(138)  and  the  first  of  (141).  Consequently  provisional  fractional 
limits  for  the  components  can  be  determined  by  (143),  and  the 
corresponding  precision  measures  by  the  last  n  of  equations  (141). 
Beyond  this  point  the  problem  is  identical  with  the  preceding 
case,  except  that  the  adjusted  limits  of  precision  must  satisfy 
(142)  in  place  of  (139). 

The  methods  developed  in  the  preceding  articles  are  entirely 
general  and  applicable  to  any  form  of  the  function  6,  but  they 
frequently  lead  to  complicated  computations.  In  the  present 
article  we  have  seen  how  the  discussion  can  be  simplified  when  the 
function  0  can  be  put  in  either  of  the  particular  forms  represented 
by  (134)  and  (135).  Many  of  the  problems  met  with  in  practice 
cannot  be  put  in  either  of  these  special  forms,  but  it  frequently 
happens  that  the  treatment  of  the  functions  representing  them 
can  be  simplified  by  a  suitable  modification  or  combination  of  the 
above  general  and  particular  methods.  The  general  ideas  under- 
lying all  discussions  of  the  necessary  precision  of  components 
have  been  discussed  above  with  sufficient  fullness  to  show  their 
nature  and  significance.  Their  application  to  particular  prob- 
lems must  be  left  to  the  ingenuity  of  the  observer  and  computer. 

81.  Numerical  Example.  —  As  an  illustration  of  the  fore- 
going methods,  suppose  that  the  electromotive  force  of  a  battery 
is  to  be  determined,  and  that  the  precision  measure  of  the  result 
is  required  to  satisfy  the  condition 

R0  =  ±  0.0012  volts,  (i) 

T-> 

within  the  limits  ±  T?!>i-e->  #o  must  lie  between  ±  0.0011  and 

=b  0.0013  volt.  Preliminary  considerations  demand  that  the 
expense  of  the  work  shall  be  as  low  as  is  consistent  with  the 
required  precision. 


158 


THE    THEORY  OF  MEASUREMENTS       [ART.  81 


The  given  conditions  are  most  likely  to  be  fulfilled  by  some 
form  of  potentiometer  method.  Suppose  that  the  arrangement 
of  apparatus  illustrated  in  Fig.  10  is  adopted  provisionally;  and, 
to  simplify  the  discussion,  suppose  that  the  various  parts  of  the 
apparatus  are  so  well  insulated  that  leakage  currents  need  not 
be  considered.  The  generality  of  the  problem  is  not  appreciably 
affected  by  the  latter  assumption  since  the  specified  condition 
can  be  easily  satisfied  in  practice  within  negligible  limits.  With 
what  precision  must  the  several  components  and  correction 
factors  be  determined  in  order  that  equation  (i)  may  be  satisfied? 


-T&Z 


FIG.  10. 


Let  V  =  e.m.f.  of  tested  battery  BI, 

Et  =  e.m.f.  of  Clark  cell  B2  at  time  of  observation, 
t    =  temperature  of  Clark  cell  at  time  of  observation, 
Ri  =  resistance  between  1  and  2, 
Rz  =  resistance  between  1  and  3, 

/  =  current  in  circuit  1,  2,  3,  B3,  1  when  the  key  K  is  open, 
5i  =  algebraic  sum  of  thermo  e.m.f.'s  in  the  circuit  1,  2,  6, 

G,  1  when  K  is  closed  to  6, 
§2  =  algebraic  sum  of  thermo  e.m.f. 's  in  the  circuit  1,  3,  a, 

G,  1  when  K  is  closed  to  a, 
Ei5  —  e.m.f.  of  Clark  cell  at  temperature  15°  C., 
a.  =  mean   temperature   coefficient   of   Clark   cell   in  the 
neighborhood  of  20°  C. 


ART.  81]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  159 

When  the  sliding  contacts  2  and  3  are  so  adjusted  that  the 
galvanometer  G  shows  no  deflection  on  closing  the  key  K  to 
either  a  or  6, 


RI  RZ 

Consequently 

F  =  (^+62)|-1-51.  (ii) 

-fi/2 

But 

(in) 


Hence 

F  =  -B16!l-a«-15)jf-1+«2f-1-81.  (iv) 

KZ  n>z 

The  resistances  RI  and  #2  are  functions  of  the  temperature;  but, 

since  they  represent  simultaneous  adjustments  with  the  cells  BI 

•p 

and  Bz  and  are  composed  of  the  same  coils,  the  ratio  ~  is  inde- 

KZ 

pendent  of  the  temperature.  Thus,  if  Rt'  and  Rt"  represent  the 
resistances  of  the  used  coils  at  t°  C.,  and  ft  is  their  temperature 
coefficient, 

RS      Ri(l+  fit)      Ri 


whatever  the  temperature  t  at  which  the  comparison  is  made. 
This  advantage  is  due  to  the  particular  method  of  connection  and 
adjustment  adopted,  and  is  by  no  means  common  to  all  forms  of 
the  potentiometer  method. 

Under  the  conditions  specified  above,  equation  (iv)  may  be 
adopted  as  the  complete  expression  for  the  discussion  of  precision. 
It  corresponds  to  equation  (120)  in  the  general  treatment  of  the 
problem.  Suppose  that  the  following  approximate  values  of  the 
components,  which  are  sufficiently  close  for  the  determination  of 
the  capabilities  of  the  method,  have  been  obtained  from  the 
normal  constants  of  the  Clark  cell  and  a  preliminary  adjustment 
of  the  apparatus  or  by  computation  from  a  known  approximate 
value  of  V: 

#15  =  1.434  volts;    a  =  0.00086; 

t  =  20°  C.;          Ri  =  1000  ohms; 

R2  =  1310  ohms;    V  =  1.1  volts. 

The  thermoelectromotive  forces  5i  and  52  are  to  some  extent 

due  to  inhomogeneity  of  the  wires  used  in  the  construction  of 

the  instruments  and  connections.     For  the  most  part,  however, 


(v) 


160  THE  THEORY  OF  MEASUREMENTS       [ART.  81 

they  arise  from  the  junctions  of  dissimilar  metals  in  the  circuits 
considered.  Suppose  that  the  resistances  R\  and  #2  are  made  of 
manganin,  the  key  K  of  brass,  and  that  the  copper  used  in  the 
galvanometer  coil  and  the  connecting  wires  is  thermoelectrically 
different.  Both  5i  and  52  would  represent  the  resultant  action 
of  at  least  six  thermo-elements  in  series.  While  these  effects  can- 
not be  accurately  specified  in  advance,  their  combined  action 
would  not  be  likely  to  be  greater  than  twenty-five  microvolts  per 
degree  difference  in  temperature  between  the  various  parts  of  the 
apparatus,  and  it  might  be  much  less  than  this.  Obviously  5i 
and  62  are  both  equal  to  zero  when  the  temperature  of  the  appa- 
ratus is  uniform  throughout. 

By  equations  (133),  article  seventy-nine,  the  correction  terms 
depending  on  thermoelectric  forces  will  be  negligible  in  compar- 
ison with  the  given  precision  measure  R0,  when  5i  and  62  satisfy 
the  conditions 

.       1     #o      1  ,     -        1     flo      1 

'l*3'vT5E         ^s'vTE' 

ddi  dd2 

In  the  present  case 

Ro  =  0.0012  volt;    q  =  2; 
dV  .     dV      Rl 

sE*--1'  and  srsr 

Consequently  the  above  conditions  become 


-  •  5^i?  .  _L  _  ±  0.00028  volt  =  ±  280  microvolts, 
3        v  2       —  1 

_L  -  ±  0.00037  volt  =  ±  370  microvolts. 
0.76 

From  the  above  discussion  of  the  possible  magnitude  of  the  thermo- 
electromotive  forces  in  the  circuits  considered,  it  is  obvious  that 
these  limits  correspond  to  temperature  differences  of  approxi- 
mately ten  degrees  between  the  various  parts  of  the  apparatus. 
Since  the  temperature  of  the  apparatus  can  be  easily  maintained 
uniform  within  five  degrees,  the  last  two  terms  in  equation  (iv) 
are  negligible  within  the  limits  of  precision  set  in  the  present 
problem.  Hence,  for  the  determination  of  the  required  precision 
of  the  remaining  components,  the  functional  relation  (iv)  may  be 
taken  in  the  form 

(vi) 


ART.  81]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  161 


By  equation  (123),  article  seventy-six,  the  primary  condition 
for  determining  the  necessary  precision  of  the  components  is 

R<?  =  144  X  10~8  =  Z>!2  +  £>22  +  D32  +  £>42  +  D<?,       (vii) 
where  dV  67  67 


67 


(viii) 


and  EI,  EZ,  E3,  E^  E$  are  the  required  probable  errors  of  EI$,  a,  t, 
Ri,  and  Rz,  respectively. 

For  the  preliminary  determination  of  the  jE"s  by  the  principle 
of  equal  effects,  equation  (127),  article  seventy-seven,  becomes 

=  ±  0.00054.     (ix) 


VN      V5 

Neglecting  all  factors  that  do  not  affect  the  differential  coefficients 
by  more  than  one  unit  in  the  second  significant  figure  and  adopt- 
ing the  approximate  values  of  the  components  given  in  (v), 

67      R!      1000      n_ 
j™-  =  p-  =  T^  =  0.76, 

d-Cns       /t2        lolU 


=-  E15a  =  -  -  0.00094, 
it/2 

=      Eu±  =  0.0011, 

it  2 


(x) 


Hence,  by  combining  (viii)  and  (ix),  or  directly  from  equations 
(128),  article  seventy-seven, 
,  0.00054 


(xi) 


Ez 

0.76 
0.00054 

—  ZEI    V7.V7UV/I  J.     VWftVj 

±n  nnnoQS 

v«*y 

(b) 
(c) 
(d) 
(e) 

5.5 
0.00054 

=  ±  0°.57  C. 
=  =b  0.49  ohm 
=  db  0.65  ohm. 

0.00094 
0.00054 

-  o.oon 

0.00054 

0.00083 

162  THE  THEORY  OF  MEASUREMENTS      [ART.  81 

In  practice  the  attainableness  of  these  limits  might  be  deter- 
mined experimentally;  but  in  the  present  case,  as  in  most  practical 
problems,  general  considerations  based  on  theory  and  previous 
experience  lead  to  equally  trustworthy  results.  In  the  first  place, 
it  is  obvious  that  the  temperature  of  the  Clark  cell  can  be  easily 
determined  closer  than  0°.6  C.  Consequently  the  limit  (c)  is  easily 
attainable  and  might  possibly  be  reduced  to  a  negligible  quantity. 

The  constants  of  the  normal  Clark  cell  are  known  well  within 
the  limits  (a)  and  (b).  But  it  requires  very  careful  treatment  of 
the  cell  to  keep  Ei6  constant  within  the  limit  (a),  and  new  cells, 
unless  they  are  set  up  with  great  care  and  skill,  are  likely  to  vary 
among  themselves  and  from  the  normal  cell  by  more  than  0.0007 
volt.  Consequently  the  limit  (a)  is  somewhat  smaller  than  is 
desirable  in  practical  work  of  the  precision  considered  in  the 
present  problem.  On  the  other  hand,  the  limit  (b)  is  very  rarely 
exceeded  by  either  old  or  new  cells  unless  they  are  very  care- 
lessly constructed  and  handled.  Hence  E2  could  probably  be 
reduced  to  the  negligible  limit. 

With  a  suitable  galvanometer,  the  nominal  values  of  the  resist- 
ances Ri  and  R%  can  be  easily  adjusted  within  the  limits  (d)  and 
(e).  But  EI  and  E5  must  be  considered  practically  as  the  pre- 
cision measures  of  R i  and  R2.  They  include  the  calibration 
errors  of  the  resistances,  the  errors  due  to  leakage  between  the 
terminals  of  individual  coils,  and  the  errors  due  to  nonuniformity 
of  temperature  as  well  as  the  errors  of  setting  of  the  contacts  2 
and  3,  Fig.  10.  The  resultant  of  these  errors  can  be  reduced 
below  the  limits  (d)  and  (e),  but  in  the  present  case  it  would  be 
convenient  to  have  somewhat  larger  limits  in  order  to  reduce  the 
expense  of  construction  and  calibration. 

Hence,  while  all  of  the  E's  given  by  equations  (xi)  are  within 
attainable  limits,  the  preliminary  consideration  of  minimum 
expense  would  be  more  likely  to  be  fulfilled  if  the  limits  (a), 
(d),  and  (e)  were  somewhat  larger.  Obviously  the  magnitude  of 
these  limits  can  be  increased  without  violating  the  primary  con- 
dition (vii)  provided  a  corresponding  decrease  in  the  magnitudes 
of  the  limits  (b)  and  (c)  is  possible. 

By  equation  (131),  article  seventy-nine,  the  separate  effects  D2 
and  DZ  will  be  simultaneously  negligible  if 
n        n        1  #o       1  0.0012 

1/2  =  DZ  = 7^  =  ~    ;=—  ^  ± 

3  Vq      3     V2 


ABT.SI]  DISCUSSION  OF  PROPOSED  MEASUREMENTS  163 

Hence,  by  equations  (132),  the  errors  of  a  and  t  will  be  negligible 

when  0.00028 

E2  =  ±  —^~  =;  ±  0.000051,  (b') 

and 

0  00028 

Ets±  mm  ^±o°-3oc.  (C') 

Since  these  limits  can  be  reached  with  much  greater  ease  than  the 
limits  (a),  (d),  and  (e),  they  may  be  adopted  as  final  specifica- 
tions and  the  corresponding  Z)'s  may  be  omitted  during  the  deter- 
mination of  new  limits  for  the  components  E15)  R1}  and  R%. 
Under  these  conditions,  equation  (ix)  becomes 


Hence  the  largest  allowable  limits  for  the  errors  of  EM,  Ri,  and 

RZ  are  0  OOOfiQ 

±      ~  =  ±  0.00091  volt,  (a') 

=  ±0.63  ohm,  (d') 


While  these  limits  cannot  be  quite  so  easily  attained  as  (b')  and 
(c'),  they  cannot  be  increased  without  violating  the  primary  con- 
dition (vii).  Consequently  they  satisfy  the  condition  of  minimum 
expense,  so  far  as  the  proposed  method  is  concerned,  and  may  be 
adopted  as  final  specifications. 

The  fractional  errors  corresponding  to  the  specified  precision 
measure  of  V  and  the  above  limiting  errors  of  the  components 

Po  =  Y  =  ±  0.0011  =  ±  0.11%, 
Pi  =  ~  =  db  0,00063  =  ±  0.063%, 
P2  =  ^  =  ±  0.059  =  =t  5.9%, 
P3  =  y  =  ±  0.015  =  db  1.5%, 
P4  =  ~  =  =fc  0.00063  =  d=  0.063%, 
P5  =  f-5  =  ±  0.00063  =  ±  0.063%. 


164  THE  THEORY  OF  MEASUREMENTS       [ART.  81 

Consequently  in  order  to  obtain  a  value  of  V  that  is  exact  within 
0.11  per  cent  by  the  proposed  method,  a  must  be  determined 
within  5.9  per  cent,  t  within  1.5  per  cent,  and  E.-&,  Ri,  and  R2)  each 
within  0.063  per  cent.  These  limits  are  all  attainable  in  practice 
under  suitable  conditions,  as  pointed  out  above.  Hence  the  pro- 
posed method  is  practicable. 

If  the  final  measurements  are  so  devised  and  executed  that  the 
above  conditions  are  fulfilled,  the  precision  of  the  result  computed 
from  them  will  be  within  the  specified  limits  and  the  expense  of 
the  work  will  be  reduced  to  the  lowest  limit  compatible  with  the 
proposed  method.  The  desired  result  might  be  obtained  at  less 
expense  by  some  other  method,  but  a  decision  on  this  point  can 
be  reached  only  by  comparing  the  precision  requirements  and 
practicability  of  various  methods  with  the  aid  of  analyses  similar 
to  the  above. 


CHAPTER  XII. 
BEST  MAGNITUDES  FOR  COMPONENTS. 

82.  Statement  of  the  Problem.  —  The  precision  of  a  derived 
quantity  depends  on  the  relative  magnitudes  and  precision  of  the 
components  from  which  it  is  computed,  as  explained  in  Chapter 
VIII.  Thus,  if  the  derived  quantity  XQ  is  given  in  terms  of  the 
components  x\,  x^  .  .  .  ,  xq  by  the  expression 

x0  =  F  (xi,  x2,  .  .  .  ,  xg),  (144) 


the  probable  error  of  XQ  is  given  by  the  expression 

EQ*  =  SSEJ  +  S22E2Z  +  •  •  •  +  Sq*Eq2,  (145) 

where  the  E's  represent  the  probable  errors  of  the  x's  with  corre- 
sponding subscripts,  and 

AF  AF  AF 

*-&  *-•&•••'«••<£•        (146) 

The  error  E,  corresponding  to  any  directly  measured  com- 
ponent, is  generally,  but  not  always,  independent  of  the  absolute 
magnitude  of  that  component  so  long  as  the  measurements  are 
made  by  the  same  method  and  apparatus.  For  example:  the 
probable  error  of  a  single  measurement  with  a  micrometer  caliper, 
graduated  to  0.01  millimeter,  is  approximately  equal  to  0.004 
millimeter,  whatever  the  magnitude  of  the  object  measured  so 
long  as  it  is  within  the  range  of  the  instrument.  Hence,  when 
the  methods  and  instruments  to  be  used  in  measuring  each  of 
the  components  are  known  in  advance,  the  probable  errors  EI, 
E2,  etc.,  can  be  determined,  at  least  approximately,  by  preliminary 
measurements  on  quantities  of  the  same  kind  as  the  components 
but  of  any  convenient  magnitude.  Under  these  conditions  the 
E's  on  the  right-hand  side  of  equation  (145)  may  be  treated  as 
known  constants,  and,  since  the  S's  are  expressible  in  terms  of 
Xi,  xz,  etc.,  by  equations  (146),  the  value  of  E0  corresponding  to 
the  given  methods  cannot  be  changed  without  a  simultaneous 
change  in  the  relative  or  absolute  magnitudes  of  the  components. 

165 


166  THE  THEORY  OF  MEASUREMENTS       [ART.  82 

Since  equation  (144)  must  always  be  fulfilled,  and  since  the 
value  of  XQ  is  usually  fixed  by  the  conditions  of  the  problem,  a 
change  in  the  magnitudes  of  the  re's  is  not  always  possible.  But 
it  frequently  happens  that  the  form  of  the  function  F  is  such  that 
the  relative  magnitudes  of  the  components  can  be  changed  through 
somewhat  wide  limits  and  still  satisfy  equation  (144).  Thus,  if 
a  cylinder  is  to  have  a  specified  volume,  it  may  be  made  long  and 
thin,  or  short  and  thick,  and  have  the  same  volume  in  either  case. 
Consequently  it  is  sometimes  possible  to  select  magnitudes  for 
the  components  that  will  give  a  minimum  value  of  E0  and  at  the 
same  time  satisfy  equation  (144). 

The  problem  before  us  may  be  briefly  stated  as  follows  :  Having 
given  definite  methods  and  apparatus  for  the  measurement  of  the 
components  of  a  derived  quantity  reo,  what  magnitudes  of  the 
components  will  give  a  minimum  value  to  the  probable  error  EQ  of 
XQ  and  at  the  same  time  satisfy  the  functional  relation  (144)? 

It  can  be  easily  seen  that  a  practical  solution  of  this  problem 
is  not  always  possible.  In  the  first  place  the  form  of  the  function 
F  may  be  such  as  to  admit  of  but  a  single  system  of  magnitudes 
of  the  components,  and  consequently  the  value  of  EQ  is  definitely 
fixed  by  equation  (145).  In  some  cases  there  are  no  real  values 
of  the  re's  that  will  satisfy  both  (144)  and  the  conditions  for  a 
minimum  of  EQ.  When  values  can  be  found  that  satisfy  the 
mathematical  conditions  they  are  not  always  attainable  in  prac- 
tice. Finally  the  probable  errors  Ei,  E2,  etc.,  may  not  be  inde- 
pendent of  the  magnitudes  of  the  corresponding  components  or 
it  may  be  impossible  to  determine  them  in  advance  of  the  final 
measurements. 

When  the  E's  are  not  independent  of  the  re's  it  sometimes 
happens  that  the  fractional  errors 

Pi  =  ?;  p*  =  ?  '•••>  p*  =  ?        (147) 

3/1  it/2  Xq 

are  constant  and  determinable  in  advance.     In  such  cases  the 
problem  may  be  solvable  by  putting  (145)  in  the  equivalent  form 

Ef  =  SfPfy?  +  SfPfxf  +!»>•+  Sq*Pq*xq*,        (148) 


expressing  the  S's  in  terms  of  the  components  by  equations  (146), 
and  determining  the  values  of  the  re's  that  will  render  (148)  a 
minimum  subject  to  the  condition  (144). 


ART.  83]     BEST  MAGNITUDES  FOR  COMPONENTS       167 

When  a  practicable  solution  of  the  problem  is  possible,  it  is 
obvious  that  the  results  thus  obtained  are  the  best  magnitudes 
that  can  be  assigned  to  the  components,  and  that  they  should 
be  adopted  as  nearly  as  possible  in  carrying  out  the  final  measure- 
ments from  which  XQ  is  to  be  computed. 

83.  General  Solutions.  —  The  general  conditions  for  a  mini- 
mum or  a  maximum  value  of  EQ2,  when  XQ  is  treated  as  a  constant 
and  the  variables  are  required  to  satisfy  the  relation  (144),  but 
are  otherwise  independent,  are 


dF 
^  A  — —  =  U, 


0) 


where  K  is  an  arbitrary  constant.  By  introducing  the  expressions 
(145)  and  (146),  transposing  and  dividing  by  two,  equations  (i) 
become 


Slgtf1.  +  S,g^+... 
o    O&1  ET  2  _j_  O    0O2  pi  2     i 

1  dx2  2  ^2 


(149) 


When  the  S's  have  been  replaced  by  x's  with  the  aid  of  equa- 
tions (146),  the  q  equations  (149),  together  withj(144),  are  theoreti- 
cally sufficient  for  the  determination  of  all  of  the  q  +  1  unknown 
quantities  Xi,  x2,  .  .  .  ,  xq,  and  K.  However,  in  some  cases  a 
practicable  solution  is  not  possible,  and  in  others  the  components 
or  their  ratios  come  out  as  the  roots  of  equations  of  the  second 
or  higher  degree.  The  zero,  infinite,  and  imaginary  roots  of  these 
equations  have  no  practical  significance  in  the  present  discussion 
and  need  not  be  considered.  Some  of  the  real  roots  correspond  to 
a  maximum,  some  to  a  minimum,  and  others  to  neither  a  maximum 
nor  a  minimum  value  of  E0Z.  In  most  cases  the  roots  that  corre- 
spond to  a  minimum  of  E02  can  be  selected  by  inspection  with  the 


168 


THE  THEORY  OF  MEASUREMENTS       [ART.  83 


aid  of  equation  (145),  but  it  is  sometimes  necessary  to  apply  the 
well-known  criteria  of  the  calculus. 

Dividing  equation  (145)  by  xQ2  and  putting 

XQ  dX2 '  q       XQ        XQ  dXq 


XQ        XQ  dXi 

gives  the  expression 

PZ  =  EI 

X02 


XQ 


(150) 


+  T*E*         (151) 


for  the  fractional  error  of  XQ.  Since  XQ  is  a  constant  in  any  given 
problem  the  maxima  and  minima  of  P02  correspond  to  the  same 
values  of  the  components  as  those  of  EQ2.  Sometimes  the  form 
of  the  function  F  is  such  that  the  expression  (151),  when  expanded 
in  terms  of  the  x's,  is  much  simpler  than  (145).  In  such  cases  it 
is  much  easier  to  determine  the  minima  of  P02  than  of  E02.  For 
this  purpose  the  equations  of  condition  (i)  may  be  put  in  the  form 


6X1 


XQ   dXi 

KdF_ 

XQ  6X2 


dx, 


(152) 


,q  XQ  dXq 

and  by  substitution  and  transposition  we  have 

dTi  dT%  dTg 

1  dxi  2  dxi     2  q  dxi 


dT< 


(153) 


When  the  components  are  required  to  satisfy  the  condition  (144) 
and  a  given  constant  value  is  assigned  to  XQ,  equations  (153)  lead 
to  exactly  the  same  results  as  equations  (149).  In  fact  either  of 
these  sets  of  equations  can  be  derived  from  the  other  by  purely 
algebraic  methods  when  the  $'s  and  T's  are  expressed  in  terms  of 
the  x's.  In  practice  one  or  the  other  of  the  sets  will  be  the  simpler, 
depending  on  the  form  of  the  function  F;  and  the  simpler  form 


ART.  83]     BEST  MAGNITUDES  FOR  COMPONENTS       169 

can  be  more  easily  derived  by  direct  methods  as  above  than  by 
algebraic  transformation. 

In  some  problems  the  magnitude  of  one  or  more  of  the  com- 
ponents in  the  function  F  can  be  varied  at  will  and  determined 
with  such  precision  that  their  probable  errors  are  negligible  in 
comparison  with  those  of  the  other  components.  Variables  that 
fulfill  these  conditions  will  be  called  free  components.  Since  any 
convenient  magnitude  can  be  assigned  to  them,  their  values  can 
always  be  so  chosen  that  the  condition  (144)  will  be  fulfilled 
whatever  the  values  of  the  other  components.  Consequently  the 
latter  components  may  be  treated  as  independent  variables  in 
determining  the  minima  of  EQ2  or  PQ2. 

Under  these  conditions  the  E's  corresponding  to  the  free  com- 
ponents can  be  placed  equal  to  zero,  and  either  E02  or  P02  can 
sometimes  be  expressed  as  a  function  of  independent  variables 
only  by  eliminating  the  free  components  from  the  S's  or  the  T's 
with  the  aid  of  equation  (144).  When  this  elimination  can  be 
effected,  the  minimum  conditions  may  be  derived  from  equations 
(149)  or  (153),  as  the  case  may  be,  by  placing  K  equal  to  zero  and 
omitting  the  equations  involving  derivatives  with  respect  to  the 
free  components.  This  is  evident  because  the  remaining  com- 
ponents are  entirely  independent,  and  consequently  the  partial 
derivatives  of  EQ2  or  P02  with  respect  to  each  of  them  must  vanish 
when  the  values  of  the  variables  correspond  to  the  maxima  or 
minima  of  these  functions.  When  the  elimination  cannot  be 
accomplished,  neither  equations  (149)  nor  (153)  will  lead  to  con- 
sistent results  and  the  problem  is  generally  insolvable. 

In  practice  it  frequently  happens  that  the  free  components  are 
factors  of  the  function  F,  and  are  not  included  in  any  other  way. 
Under  these  conditions  they  do  not  occur  in  the  T's  corresponding 
to  the  remaining  components,  since  the  form  of  equations  (150) 
is  such  that  they  are  automatically  eliminated.  Consequently, 
in  this  case,  the  conditions  for  a  minimum  are  given  at  once  by 
equations  (153)  when  K  is  taken  equal  to  zero,  since  the  derivatives 
with  respect  to  the  free  components  all  vanish  and  the  correspond- 
ing E's  are  negligible.  It  is  scarcely  necessary  to  point  out  that 
the  remarks  in  the  paragraph  following  equations  (149),  except 
for  obvious  changes  in  notation,  apply  with  equal  rigor  to  equa- 
tions (153),  whether  K  is  zero  or  finite.  The  values  of  the  x's 
derived  from  these  equations  should  never  be  assumed  to  corre- 
spond to  the  minima  of  P02  without  further  investigation. 


170  THE  THEORY  OF  MEASUREMENTS       [ART.  84 

84.  Special  Cases.  —  Suppose  that  the  relation  between  the 
derived  quantity  XQ  and  the  measured  components  xi,  #2,  and  xs 
is  given  in  the  form 

XQ  =  ax?*  +  bxj1*  +  cxj1*,  (ii) 

where  a,  b,  c,  and  the  n's  are  constants.  If  the  probable  errors 
Eit  Ez,  and  E3  of  the  x's  with  corresponding  subscripts  are  known, 
and  independent  of  the  magnitude  of  the  components,  what  mag- 
nitudes of  the  components  will  give  the  least  possible  value  to  the 
probable  error  E0  of  XQ? 
By  equations  (146), 

Si  =  arnxi^-V;     S2  =  bn&^'-V;    Ss  =  c/W^-D.       (iii) 
Consequently 

dSi  ,  i\        („       <>\  ^$2          rv  ^$3  _. 

—  -«»!(„,  -I)**-*;  ._  =0;     —  =  0, 


Substituting  these  results  in  equations  (149)  and  dividing  the 
first  equation  by  Si,  the  second  by  $2,  and  the  third  by  SS)  the 
conditions  for  a  minimum  value  of  EQ2  become 

Efari!  (m  -  1)  xi<*-*>  =  K, 


Dividing  the  second  and  third  of  these  equations  by  the  first 
and  transposing  the  coefficients  to  the  second  member  gives  the 
ratios  of  the  components  in  the  form 

x2(n^-2)  =  EJani  (ni-l) 

T,(tti-2)   ~~    EL2Jmn  (nn  •-  IV 


(HI  -  1) 


~ 

(ns- 

These  two  equations  together  with  (ii)  are  theoretically  sufficient 
for  the  determination  of  the  best  magnitudes  for  the  three  com- 
ponents xij  Xzj  and  x$]  but  it  can  be  easily  seen,  from  the  form  of 
the  equations,  that  a  solution  is  not  practicable  for  all  possible 
values  of  the  n's. 


ART.  84]    BEST  MAGNITUDES  FOR  COMPONENTS        171 

For  example,  if  the  n's  are  all  equal  to  unity,  the  ratios  of  the 
components  given  by   (iv)  are  both  indeterminate,  each  being 

equal  to  ^-     Consequently  the  problem  has  no  solution  in  this 

case.  This  conclusion  might  have  been  reached  at  once  by 
inspecting  the  value  of  EQ2  given  by  equation  (145),  when  the  S's 
are  expressed  in  terms  of  the  components.  Thus,  placing  the  n's 
equal  to  unity  in  equations  (iii)  and  substituting  the  results  in 
(145),  we  find 


Since  E<?  is  independent  of  the  x's  it  can  have  no  maxima  or 
minima  with  respect  to  the  components. 

When  each  of  the  n's  equals  two,  equations  (iv)  are  inde- 
pendent of  the  x's,  and  consequently  the  problem  is  not  solvable. 
In  this  case  (ii)  becomes 

XQ  = 

and  (145)  reduces  to 
E02  =  4 

Since  these  equations  differ  only  in  the  values  of  the  constant 
coefficients  of  the  x's,  no  magnitudes  can  be  assigned  to  the  com- 
ponents that  will  give  a  minimum  value  to  EQ2,  and  at  the  same 
time  satisfy  the  equation  for  XQ. 

If  each  of  the  n's  is  placed  equal  to  three,  equation  (ii)  takes 
the  form 

XQ  =  ax^  +  bx2*  +  c#33,  (v) 

and  equations  (iv)  become 

Xt~bEf' 

(iv') 

C#32 

In  this  case  the  problem  can  be  easily  solved  when  the  numerical 
values  of  the  coefficients  and  the  E's  are  known.  As  a  very 
simple  illustration,  suppose  that 

7  -f  J  77T       -TGI  ~Ij1          —     XT' 

a  =  o  =  c  =  1,    and    J^i  =  &2  —  MS  —  &, 
then,  by  (iv')  and  (v), 


and,  by  (145)  and  (iii), 


172  THE  THEORY  OF  MEASUREMENTS       [ART.  84 

Since  a  decrease  in  the  magnitude  of  one  of  the  x's  involves  an 
increase  in  that  of  one  or  both  of  the  others,  in  order  to  satisfy 
equation  (v),  and  since  the  fourth  power  of  a  quantity  varies 
more  rapidly  than  the  third,  it  is  obvious  that  the  minimum 
value  of  E02  will  occur  when  the  x's  are  all  equal.  Consequently 
the  above  solution  corresponds  to  a  minimum  of  E02. 

It  can  be  easily  seen  that  there  are  many  other  cases  in  which 
equations  (ii)  and  (iv)  can  be  solved,  and  also  some  others  in 
which  no  solution  is  possible.  The  extension  of  the  problem  to 
functions  in  the  same  form  as  equation  (ii),  but  containing  any 
number  of  similar  terms,  involves  only  the  addition  of  one  equa- 
tion in  the  form  of  (iv)  for  each  added  component.  Obviously 
these  equations  hold  for  negative  as  well  as  positive  values  of  the 
coefficients  and  exponents  of  the  x's. 

As  a  second  example,  consider  the  functional  relation 

x0  =  axini  X  xf*.  (vi) 


In  this  case  the  solution  is  more  easily  effected  by  the  second 
method  given  in  the  preceding  article.     By  equations  (150) 


Consequently 

and  equations  (153)  reduce  to  the  simple  form 

^ES=-K;     %Ef  =  -K,          ;  (viii) 

where  EI  and  E2  are  the  known  constant  probable  errors  of  Xi  and 
#2.     Eliminating  K,  we  have 


Consequently  the  problem  is  always  solvable  when  n\  and  n2 
have  the  same  sign.  When  they  have  different  signs  the  solu- 
tion is  imaginary.  Hence  there  are  no  best  magnitudes  for  the 
components  when  the  derived  quantity  is  given  as  the  ratio  of 
two  measured  quantities. 


ART.  85]     BEST  MAGNITUDES  FOR  COMPONENTS       173 

The  extension  of  this  solution  to  functions  involving  any  num- 
ber of  factors  is  obvious.  When  the  exponents  of  all  of  the 
factors  have  the  same  sign  the  problem  is  always  solvable  but 
the  best  magnitudes  thus  found  may  not  be  attainable  in  practice. 
If  part  of  the  exponents  are  positive  and  others  are  negative  the 
solution  is  imaginary. 

85.  Practical  Examples. 
I. 

In  many  experiments  the  desired  result  depends  directly  upon 
the  determination  of  the  quantity  of  heat  generated  by  an  electric 
current  in  passing  through  a  resistance  coil.  Let  I  represent  the 
current  intensity  and  E  the  fall  of  potential  between  the  terminals 
of  the  coil.  Then  the  quantity  of  heat  H  developed  in  t  seconds 
may  be  computed  by  the  relation 

JH  =  TEt, 

where  J  represents  the  mechanical  equivalent  of  heat.     If  H  is 
measured  in  calories,  I  in  amperes,  E  in  volts,  and  t  in  seconds, 

•y  is  equal  to  0.239  calorie  per  Joule  and  the  above  relation  becomes 

H  =  0.239  •  lEt.  (ix) 

Suppose  that  the  conditions  of  the  problem  in  hand  are  such 
that  H  should  be  made  approximately  equal  to  1000  calories. 
Since  the  resistance  of  the  heating  coil  is  not  specified  it  can  be  so 
chosen  that  7  and  E  may  have  any  convenient  values  that  satisfy 
the  relation  (ix)  when  H  has  the  above  value.  Obviously  t  can 
be  varied  at  will,  by  changing  the  time  of  run,  and  (ix)  will  not 
be  violated  if  suitable  values  are  assigned  to  /  and  E.  If  the 
instruments  available  for  measuring  /,  E,  and  t  are  an  ammeter 
graduated  to  tenths  of  an  ampere,  a  voltmeter  graduated  to 
tenths  of  a  volt,  and  a  common  watch  with  a  seconds  hand,  what 
are  the  best  magnitudes  that  can  be  assigned  to  the  components, 
i.e.,  what  magnitudes  of  /,  E,  and  t  will  give  the  computed  H 
with  the  least  probable  error? 

By  comparing  equations  (ix)  and  (vi),  it  is  easy  to  see  that 
the  present  problem  is  an  application  of  the  second  special  case 
worked  out  in  the  preceding  article  when  a  third  variable  factor 
Z3n3  is  annexed  to  (vi).  H  corresponds  "to  x0)  I  to  Xi,  E  to  xz,  t  to 
#3/and  all  of  the  n's  in  (vi)  are  equal  to  unity.  Consequently 


174  THE  THEORY  OF  MEASUREMENTS       [ART.  85 

the  solution  can  be  derived  at  once  from  three  equations  in  the 
form  of  (viii)  if  suitable  values  can  be  assigned  to  the  probable 
errors  of  the  components. 

With  the  available  instruments,  the  probable  errors  Ei}  Ee,  and 
Et  of  /,  E,  and  t,  respectively,  will  be  practically  independent  of 
the  magnitude  of  the  measured  quantities  so  long  as  the  range 
of  the  instruments  is  not  exceeded.  Under  the  conditions  that 
usually  prevail  in  such  observations  the  following  precision  may 
be  attained  with  reasonable  care: 

Et  =  0.05  ampere;    Ee  =  0.05  volt;    Et  =  1  second. 

The  conditions  for  a  minimum  value  of  the  probable  error  E0 
of  H  can  be  derived  by  exactly  the  same  method  that  was  used 
in  obtaining  equations  (viii),  or  these  equations  may  be  used  at 
once  with  proper  substitutions  as  outlined  above.  Consequently 
the  best  magnitudes  for  the  components  are  given  by  the  simul- 
taneous solution  of  (ix)  and  the  following  three  equations, 

^2_       K.    ^_       „     E? 
~P  =    ~K>    ~W~    ~K>    ~P  = 

Eliminating  K  and  substituting  the  numerical  values  of  the 
probable  errors  we  have 

E_Ee_.  l_Et_ 

I  ~  E<~  L>  I~  Ei~ 

Consequently 

E  =  I    and  t  =  20  •  /.  (x) 

Substituting  these  results  and  the  numerical  value  of  H  in  (ix) 
we  have 

1000  =  0.239  X  20  X  /3, 
and  hence 

I  =  5.94  amperes 

is  the  best  magnitude  to  assign  to  the  current  strength  under  the 
given  conditions.  The  corresponding  magnitudes  for  the  electro- 
motive force  and  time  found  by  (x)  are 

E  =  5.94  volts  and  t  =  119  seconds. 

If  the  above  values  of  the  components  and  their  probable  errors 
are  substituted  in  equation  (151),  the  fractional  error  of  H  comes 
out 


ART.  85]    BEST  MAGNITUDES  FOR  COMPONENTS       175 

and  the  probable  error  of  H  is  given  by  the  relation 
EQ  =  1000  Po  =±15  calories. 

If  any  other  magnitudes  for  the  components,  that  satisfy  equa- 
tion (ix),  are  used  in  place  of  the  above  in  (151),  the  computed 
value  of  E0  will  be  greater  than  fifteen  calories.  Consequently 
the  above  solution  corresponds  to  a  minimum  value  of  EQ. 

In  order  to  fulfill  the  above  conditions  the  resistance  of  the 
heating  coil  must  be  so  chosen  as  to  satisfy  the  relation 

*-£ 

Since  our  solution  calls  for  numerically  equal  values  of  I  and  E, 
the  resistance  R  must  be  made  equal  to  one  ohm. 

It  can  be  easily  seen  that  small  variations  in  the  values  of  the 
components  will  produce  no  appreciable  effect  on  the  probable 
error  of  H,  ^ince  the  numerical  value  of  E0  is  never  expressed  by 
more  than  two  significant  figures.  Consequently  the  foregoing 
discussion  leads  to  the  following  practical  suggestions  regarding 
the  conduct  of  the  experiment.  The  heating  coil  should  be  so 
constructed  that  the  heat  developed  in  the  leads  is  negligible  in 
comparison  with  that  developed  between  the  terminals  of  the 
voltmeter.  The  resistance  of  the  coil  should  be  one  ohm.  The 
current  strength  should  be  adjusted  to  approximately  six  amperes 
and  allowed  to  flow  continuously  for  about  two  minutes.  Under 
these  conditions  the  difference  in  potential  between  the  terminals 
of  the  coil  will  be  about  six  volts.  The  conditions  under  which 
7,  E,  and  t  are  observed  should  be  so  chosen  that  the  probable 
errors  specified  above  are  not  exceeded. 

If  the  above  suggestions  are  carried  out  in  practice  the  value 
of  H  computed  from  the  observed  values  of  /,  E,  and  t  by  equa- 
tion (ix)  will  be  approximately  1000  calories,  and  its  probable 
error  will  be  about  fifteen  calories.  A  more  precise  result  than 
this  cannot  be  obtained  with  the  given  instruments  unless  the 
probable  errors  of  7,  E,  and  t  can  be  materially  decreased  by 
modifying  the  conditions  and  methods  of  observation. 

II. 

A  partial  discussion  of  the  problem  of  finding  the  best  magni- 
tudes for  the  components  involved  in  the  measurement  of  the 
strength  of  an  electric  current  with  a  tangent  galvanometer  may 


176  THE   THEORY  OF  MEASUREMENTS       [ART.  85 

be  found  in  many  laboratory  manuals  and  textbooks.  Such  dis- 
cussions are  usually  confined  to  a  consideration  of  the  error  in  the 
computed  current  strength  due  to  a  given  error  in  the  observed 
deflection.  On  the  assumption,  tacit  or  expressed,  that  the  effects 
of  the  errors  of  all  other  components  are  negligible  it  is  proved 
that  the  effect  of  the  deflection  error  is  a  minimum  when  the 
deflection  is  about  forty-five  degrees.  Although  the  tangent  gal- 
vanometer is  now  seldom  used  in  practice  it  provides  an  instructive 
example  in  the  calculation  of  best  magnitudes  since  the  general 
bearings  of  the  problem  are  already  familiar  to  most  students. 

In  order  to  avoid  unnecessary  complications,  consider  a  simple 
form  of  instrument  with  a  compass  needle  whose  position  is 
observed  directly  on  a  circle  graduated  in  degrees.  Suppose  that 
the  needle  is  pivoted  at  the  center  of  a  single  coil  of  N  turns  of 
wire,  and  R  centimeters  mean  radius.  Under  these  conditions  the 
current  strength  I  is  connected  with  the  observed  deflection  (f>  by 
the  relation 


where  H  is  the  horizontal  intensity  of  a  uniform  external  magnetic 
field  parallel  to  the  plane  of  the  coil.  In  practice  the  plane  of  the 
coil  is  usually  placed  parallel  to  the  magnetic  meridian  and  H 
is  taken  equal  to  the  horizontal  component  of  the  earth's  mag- 
netism. 

N  is  an  observed  component  but  it  can  be  so  precisely  deter- 
mined by  direct  counting,  during  the  construction  of  the  coil, 
that  its  error  may  be  considered  negligible  in  comparison  with 
those  of  the  other  components.  Furthermore  it  can  be  given  any 
desired  value  when  an  instrument  is  designed  to  meet  special 
needs,  and  a  choice  among  a  number  of  different  values  is  possi- 
ble in  most  completed  instruments.  Consequently  the  quantity 

x—  TT  may  be  treated  as  a  free  component,  represented  by  A,  and 
the  expression  for  the  current  strength  may  be  written  in  the 
form  7  =  A#£.tan0.  (xi) 

Comparing  this  expression  with  the  general  equation  (144)  we 
note  that  /  corresponds  to  x0)  H  to  x\,  R  to  x2,  and  0  to  z3. 

Since  A  is  free,  the  components  H,  R,  and  </>  are  entirely  inde- 
pendent; and  any  convenient  magnitudes  can  be  made  to  satisfy 


ART.  85]    BEST  MAGNITUDES  FOR  COMPONENTS       177 

(xi)  by  suitably  choosing  the  number  of  turns  in  the  coil.  Con- 
sequently, as  pointed  out  in  article  eighty-three  with  respect  to 
functions  containing  a  free  component  as  a  factor,  the  conditions 
for  a  minimum  probable  error  of  /  are  given  by  equations  (153) 
with  K  placed  equal  to  zero.  By  making  the  above  substitutions 
for  the  x's  in  equations  (150)  and  performing  the  differentiations 
we  have 

I/'  7?'  oi-r»    O  ^  *  V^^X 

11  /L  bill  £  cp 

Consequently 

0/77  -i  z\nn  H^TI 

ol  i  1     .      o  J.  2        f\        OJ.  3       ~ 

dH=   ~H~2'     dH        ;     ~dH=    ' 

*^/T7  *\  ATT  "I  fk  T7 

?£l  —  n     ^ 2  _  _  L  •    ^ 3  _  n. 

dR  ~  dR          R2'     dR  ' 

dTi  ^  =  n-     dTz=      4cos2<?i> 

d0  60   "  60  sin2  2  0 ' 

and,  if  the  probable  errors  of  H,  R,  and  0  are  represented  by  E\9 
EZ,  and  #3,  respectively,  equations  (153)  become 


If  EI  and  E2  could  be  made  negligible,  as  is  tacitly  assumed  in 
most  discussions  of  the  present  problem,  the  first  two  of  equations 
(xiii)  would  be  satisfied  whatever  the  values  of  H  and  R.  Conse- 
quently these  components  would  be  free  and  0  would  be  the  only 
independent  variable  involved  in  equation  (xi).  Under  these 
conditions  the  minimum  value  of  the  probable  error  of  7  corre- 
sponds to  the  value  of  0  derived  from  the  third  of  equations  (xiii). 
The  general  solution  of  this  equation  is 

0=  (2n-l)|> 

where  n  represents  any  integer.  But,  since  values  of  0  greater 
than  I  are  not  attainable  in  practice,  n  must  be  taken  equal  to 

unity  in  the  present  case  and  consequently  the  best  magnitude 
for  the  deflection  is  forty-five  degrees.  It  is  obvious  that  (xi) 
can  always  be  satisfied  when  /  has  any  given  value,  and  0  is 
equal  to  forty-five  degrees  by  suitably  choosing  the  values  of  the 
free  components  2V,  H,  and  R. 


178  THE   THEORY  OF  MEASUREMENTS       [ART.  85 

If  the  fractional  error  of  /  is  represented  by  P0  and  the  T's 
given  by  equations  (xii)  are  substituted  in  (151), 


H2    '    R2    '  sin2  20 
Pi2  +  P22  +  Pa2, 


(xiv) 

=  Pi2  +  P22  +  P32, 
where 


2  •• 

=       :         and 


are  the  separate  effects  of  the  probable  errors  E\,  EZ,  and  E3) 
respectively.  If  both  ends  of  the  needle  are  read  with  direct  and 
reversed  current  so  that  0  represents  the  mean  of  four  observa- 
tions, EZ  should  not  exceed  0°.025  or  0.00044  radians,  and  it  might 
be  made  less  than  this  with  sufficient  care.  Consequently,  when 
<j>  is  equal  to  forty-five  degrees, 

P3  =  0.00088. 

By  an  argument  similar  to  that  given  in  article  seventy-nine  it  can 
be  proved  that  PI  and  P2  will  be  simultaneously  negligible  when 
they  satisfy  the  condition 

pl  =  P2  =  i  A  =  0.00021. 
3V2 

Hence,  in  order  that  the  effects  of  E\  and  E%  may  be  negligible  in 
comparison  with  that  of  E3,  H  and  R  must  be  determined  within 
about  two  one-hundredths  of  one  per  cent. 

With  an  instrument  of  the  type  considered  it  would  seldom  be 
possible  and  never  worth  while  to  determine  H  and  R  with  the 
precision  necessary  to  fulfill  the  above  condition.  In  common 
practice  E\  and  E2  are  generally  far  above  the  negligible  limit 
and  it  would  be  necessary  to  make  both  H  and  R  equal  to  infinity 
in  order  to  satisfy  the  first  two  of  the  minimum  conditions  (xiii). 
Hence  there  is  no  practically  attainable  minimum  value  of  P0. 
This  conclusion  can  also  be  derived  directly  by  inspection  of 
equation  (xiv).  P02  decreases  uniformly  as  H  and  R  are  increased, 
and  becomes  equal  to  Ps2  when  they  reach  infinity. 

Although  a  minimum  value  of  P0  is  not  attainable,  the  fore- 
going discussion  leads  to  some  practical  suggestions  regarding 
the  design  and  use  of  the  tangent  galvanometer.  For  any  given 
values  of  E\,  E2,  and  E3,  the  minimum  value  of  PS  occurs  when  <j> 
is  equal  to  forty-five  degrees.  Also  PI  and  P%  decrease  as  H  and 
R  increase.  Consequently  the  directive  force  H  and  the  radius 


ART.  85]      BEST  MAGNITUDES  FOR  COMPONENTS      179 

of  the  coil  R  should  be  made  as  large  as  is  consistent  with  the 
conditions  under  which  the  instrument  is  to  be  used,  and  the 
number  of  turns  N  in  the  coil  should  be  so  chosen  that  the  observed 
deflection  will  be  about  forty-five  degrees. 

The  practical  limit  to  the  magnitude  of  R  is  generally  set  by  a 
consideration  of  the  cost  and  convenient  size  of  the  instrument. 
Moreover  when  R  is  increased  N  must  be  increased  in  like  ratio 
in  order  to  satisfy  the  fundamental  relation  (xi)  without  altering 
the  observed  deflection  or  decreasing  the  value  of  H.  There 
is  an  indefinite  limit  beyond  which  N  cannot  be  increased  with- 
out introducing  the  chance  of  error  in  counting  and  greatly  in- 
creasing the  difficulty  of  determining  the  exact  magnitude  of  R. 
Above  this  limit  E2  is  approximately  proportional  to  R,  and,  as 
can  be  easily  seen  by  equation  (xiv),  there  is  no  advantage  to 
be  gamed  by  a  further  increase  in  the  magnitude  of  R. 

H  can  be  varied  by  suitably  placed  permanent  magnets,  but 
it  is  difficult  to  maintain  strong  magnetic  fields  uniform  and  con- 
stant within  the  required  limits.  Even  under  the  most  favorable 
conditions,  the  exact  determination  of  H  is  very  tedious  and 
involves  relatively  large  errors.  Consequently  Pi2  is  likely  to  be 
the  largest  of  the  three  terms  on  the  right-hand  side  of  equation 
(xiv).  Under  suitable  conditions  it  can  be  reduced  in  magnitude 
by  increasing  H  to  the  limit  at  which  the  value  of  EI  begins  to 
increase.  However,  such  a  procedure  involves  an  increased  value 
of  N  in  order  to  satisfy  equation  (xi),  and  consequently  it  may 
cause  an  increase  in  E2  owing  to  the  relation  between  N  and  R 
pointed  out  in  the  preceding  paragraph.  In  such  a  case  the  gain 
in  precision  due  to  a  decreased  value  of  PI  would  be  nearly  bal- 
anced by  an  increased  value  of  P%. 

In  common  practice  the  instrument  is  so  adjusted  that  H  is 
equal  to  the  horizontal  component  of  the  earth's  magnetic  field 
at  the  time  and  place  of  observation.  Unless  H  is  very  carefully 
determined  at  the  exact  location  of  the  instrument,  EI  is  likely 

to  be  as  large  as  0.005  ~5£    and,  since  the  order  of  magnitude 

Cat, 


of  H  is  about  0.2  ^r ,  -Pi  will  be  approximately  equal  to  0.025. 

cm 

Hence  both  P2  and  P3  will  be  negligible  in  comparison  with  PI  if 
they  satisfy  the  relation 

P2  =  P3  =  -  •  ^j=  =  0.0059. 
"3    V2 


180  THE  THEORY  OF  MEASUREMENTS       [ART.  85 

Under  ordinary  conditions  R  and  <£  can  be  easily  determined  within 
the  above  limit.     Consequently,  in  the  supposed  case, 
PO  =•  PI  =  2.5  per  cent, 

and  it  would  be  useless  to  attempt  an  improvement  in  precision 
by  adjusting  the  values  of  N,  R}  and  <£.  With  sufficient  care  in 
determining  H,  PI  can  be  reduced  to  such  an  extent  that  it  be- 
comes worth  while  to  carry  out  the  suggestions  regarding  the 
design  and  use  of  the  instrument  given  by  the  foregoing  theory. 
But  when  the  value  of  H  is  assumed  from  measurements  made  in 
a  neighboring  location  or  is  taken  from  tables  or  charts  the  per- 
centage error  of  /  will  be  nearly  equal  to  that  of  H  regardless  of 
the  adopted  values  of  R  and  <£.  Under  such  conditions  PQ  can- 
not be  exactly  determined  but  it  will  seldom  be  less  than  two  or 
three  per  cent  of  the  measured  magnitude  of  I. 

The  above  problem  has  been  discussed  somewhat  in  detail  in 
order  to  illustrate  the  inconsistent  results  that  are  likely  to  be 
obtained  in  determining  best  magnitudes  when  the  effects  of  the 
errors  of  some  of  the  components  are  neglected.  It  is  never 
safe  to  assume  that  the  error  of  a  component  is  negligible  until 
its  effect  has  been  compared  with  that  of  the  errors  of  the  other 
components. 

III. 

Figure  eleven  is  a  diagram  of  the  apparatus  and  connections 
commonly  used  in  determining  the  internal  resistance  of  a  bat- 
tery by  the  condenser  method.  G  is  a  ballistic  galvanometer, 
C  a  condenser,  R  a  known  resistance,  KI  a  charge  and  discharge 
key,  Kz  a  plug  or  mercury  key,  and  B  a  battery  to  be  tested. 

Let  Xi  represent  the  ballistic  throw  of  the  galvanometer  when 
the  condenser  is  charged  and  discharged  with  the  key  K2  open, 
and  xz  the  corresponding  throw  when  K2  is  closed.  Then  the 
internal  resistance  RQ  of  the  battery  may  be  computed  by  the 
relation 

Ro  =  R^L^l.  (XV) 

Under  ordinary  conditions  the  probable  errors  of  x\  and  x^ 
cannot  be  made  much  less  than  one-half  of  one  per  cent  of  the 
observed  throws  when  a  telescope,  mirror,  and  scale  are  used.  On 
the  other  hand  the  probable  error  of  R  should  not  exceed  one-tenth 
of  one  per  cent  if  a  suitably  calibrated  resistance  is  used  and  the 


ART.  85]   BEST  MAGNITUDES  FOR  COMPONENTS         181 


connections  are  carefully  made.  When  these  conditions  are  ful- 
filled, it  can  be  easily  proved  that  the  effect  of  the  error  of  R  is 
negligible  in  comparison  with  that  of  the  errors  of  Zi  and  x2. 
Furthermore  any  convenient  value  can  be  assigned  to  R,  such 


<T2  R 

"!L-A/WVW\AAAA/ 


B 
FIG.  11. 

that  (xv)  will  be  satisfied  whatever  the  values  of  Xi  and  #2.  Con- 
sequently R  may  be  treated  as  a  free  component  and  the  throws 
Xi  and  xz  as  independent  variables. 

For  the  purpose  of  determining  the  magnitudes  of  the  com- 
ponents R,  xij  and  xz  that  correspond  to  a  minimum  value  of  the 
fractional  error  P0  of  RQ,  we  have  by  equations  (150)  and  (xv) 


Consequently 


-  X2) 


(xvi) 


Since  x\  and  x2  are  independent,  K  must  be  taken  equal  to  zero 
in  the  minimum  conditions  (153).  Hence,  dividing  the  first  two 
equations  by  Ti}  we  have 

1  xi          1 


1 


E,2-^ 


=  o, 

=  0, 


(x,-xz)2  x2   x22(xl-xz)2 

where  EI  and  E2  are  the  probable  errors  of  x\  and  £2,  respectively. 


182  THE  THEORY  OF  MEASUREMENTS       [ART.  85 

Multiply  each  of  these  equations  by  --  1  ^  2        and  they  as- 
sume the  simple  form 

+-»   •     *» 


Since  #i2  and  Ez2  are  always  positive,  it  is  obvious  that  there 
are  no  values  of  Xi  and  x%  that  will  satisfy  both  of  these  equations 
at  the  same  time.  Hence,  when  Xi  and  xz  can  be  varied  inde- 
pendently, they  cannot  be  so  chosen  that  the  fractional  error  P0 
will  be  a  minimum.  However,  if  Xz  is  kept  constant  at  any  as- 
signed value,  PO  will  pass  through  a  minimum  when  Xi  satisfies 
equation  (a).  On  the  other  hand  if  any  constant  value  is  assigned 
to  Xi  the  minima  and  maxima  of  P0  will  correspond  to  the  roots 
of  equation  (b). 

In  practice  x\  is  the  throw  of  the  galvanometer  needle  due  to 
the  electromotive  force  of  the  battery  when  on  open  circuit;  and 
it  is  very  nearly  constant,  during  a  series  of  observations,  when 
suitable  precautions  are  taken  to  avoid  the  effects  of  polariza- 
tion. Both  Xi  and  Xz  can  be  varied  by  changing  the  capacity 
of  the  condenser  or  the  sensitiveness  of  the  galvanometer,  but 
their  ratio  depends  only  on  the  ratio  of  R0  to  R.  Consequently, 
if  any  convenient  magnitude  is  assigned  to  Xi,  the  root  of  equa- 
tion (b)  that  corresponds  to  a  minimum  value  of  PO  gives  the 
best  magnitude  for  the  component  Xz. 

Since  x\  and  x2  are  similar  quantities,  determined  with  the  same 
instruments  and  under  the  same  conditions,  E\  is  generally  equal 

to  EZ.     Hence,  if  we  replace  the  ratio  --  by  y,  equation  (b)  be- 

^-2^-1  =  0^  (b') 

The  only  real  root  of  this  equation  is 

y  =  2.2056. 
By  equations  (151)  and  (xvi) 


Putting  El  =  Ez  =  E     and     -  =  y, 

Xz 

Pl  =     y*  +  y* 
E*       x2-l2' 


ART.  86]     BEST  MAGNITUDES  FOR  COMPONENTS       183 

Since  Xi  is  necessarily  greater  than  x2,  y  cannot  be  less  than  unity. 

P  2 
Under  this  condition  it  can  be  easily  proved  by  trial  that  -== 

& 

approaches  a  minimum  as  y  approaches  the  value  given  above, 
provided  any  constant  value  is  assigned  to  x\. 
Equation  (xv)  may  be  put  in  the  form 
RQ  =  R(y-  1), 

and,  by  introducing  the  value  of  y  given  by  the  minimum  condi- 
tion (b')>  we  have 

R  =  0.83  R0. 

Consequently  the  greatest  attainable  precision  in  the  determina- 
tion of  RQ  will  be  obtained  when  R  is  made  equal  to  about  eighty 
three  per  cent  of  RQ.  If  R  is  adjusted  to  this  value  Xi  and  x%  will 
satisfy  equation  (b),  whatever  the  magnitude  of  the  capacity  used, 
provided  the  observations  are  so  made  that  E\  and  E%  are  equal. 

When  the  internal  resistance  of  the  battery  is  very  low  it  is 
sometimes  impracticable  to  fulfill  the  above  theoretical  conditions 
because  the  errors  due  to  polarization  are  likely  to  more  than  off- 
set the  gain  in  precision  corresponding  to  the  theoretically  best 
magnitudes  of  the  components.  In  such  cases  a  high  degree  of 
precision  is  not  attainable,  but  it  is  generally  advisable  to  make  R 
considerably  larger  than  RQ  in  order  to  reduce  polarization  errors. 

86.  Sensitiveness  of  Methods  and  Instruments.  —  The  pre- 
cision attainable  in  the  determination  of  directly  measured  com- 
ponents depends  very  largely  on  the  sensitiveness  of  indicating 
instruments  and  on  the  methods  of  adjustment  and  observation. 
The  design  and  construction  of  an  instrument  fixes  its  intrinsic 
sensitiveness;  but  its  effective  sensitiveness,  when  used  as  an  indi- 
cating device,  depends  on  the  circumstances  under  which  it  is  used 
and  is  frequently  a  function  of  the  magnitudes  of  measured  quan- 
tities and  other  determining  factors.  Thus;  the  intrinsic  sensi- 
tiveness of  a  galvanometer  is  determined  by  the  number  of 
windings  in  the  coils,  the  moment  of  the  directive  couple,  and 
various  other  factors  that  enter  into  its  design  and  construction. 
On  the  other  hand  its  effective  sensitiveness  as  an  indicator  in  a 
Wheatstone  Bridge  is  a  function  of  the  resistances  in  the  various 
arms  of  the  bridge  and  the  electromotive  force  of  the  battery 
used.  An  increase  in  the  intrinsic  sensitiveness  of  an  instrument 
may  cause  an  increase  or  a  decrease  in  its  effective  sensitiveness, 


184  THE   THEORY  OF  MEASUREMENTS       [ART.  86 

depending  on  the  nature  of  the  corresponding  modification  in 
design  and  the  circumstances  under  which  the  instrument  is 
used. 

By  a  suitable  choice  of  the  magnitudes  of  observed  components 
and  other  determining  factors  it  is  sometimes  possible  to  increase 
the  effective  sensitiveness  of  indicating  instruments  and  hence 
also  the  precision  of  the  measurements.  On  the  other  hand, 
as  pointed  out  in  Chapter  XI,  the  precision  of  measurements 
should  not  be  greater  than  that  demanded  by  the  use  to  which 
they  are  to  be  put.  In  all  cases  the  effective  sensitiveness  of 
instruments  and  methods  should  be  adjusted  to  give  a  result 
definitely  within  the  required  precision  limits  determined  as  in 
Chapter  XI.  Consequently  the  best  magnitudes  for  the  quan- 
tities that  determine  the  effective  sensitiveness  are  those  that 
will  give  the  required  precision  with  the  least  labor  and  expense. 
The  methods  by  which  such  magnitudes  can  be  determined  depend 
largely  on  the  nature  of  the  problem  in  hand,  and  a  general  treat- 
ment of  them  is  quite  beyond  the  scope  of  the  present  treatise. 
Each  separate  case  demands  a  somewhat  detailed  discussion  of 
the  theory  and  practice  of  the  proposed  measurements  and  only 
a  single  example  can  be  given  here  for  the  purpose  of  illustration. 

Since  the  potentiometer  method  of  comparing  electromotive 
forces  has  been  quite  fully  discussed  in  article  eighty-one,  it  will 
be  taken  as  a  basis  for  the  illustration  and  we  will  proceed  to  find 
the  relation  between  the  effective  sensitiveness  of  the  galvanom- 
eter and  the  various  resistances  and  electromotive  forces  involved. 
Since  the  directly  observed  components  in  this  method  are  the 
resistances  R\  and  R%,  the  effective  sensitiveness  is  equal  to  the 
galvanometer  deflection  corresponding  to  a  unit  fractional  devia- 
tion of  Ri  or  Rz  from  the  condition  of  balance. 

From  the  discussion  given  in  article  eighty-one  it  is  evident  that 
the  potentiometer  method  could  be  carried  out  with  any  conven- 
ient values  of  the  resistances  R\  and  R2  provided  they  are  so  ad- 

7-» 

justed  that  the  ratio  -—  satisfies  equation  (ii)  in  the  cited  article. 
tiz 

The  absolute  magnitudes  of  these  resistances  depend  on  the  electro- 
motive force  of  the  battery  J53  and  the  total  resistance  of  the  cir- 
cuit 1,  2,  3,  B3,  1  in  Fig.  10.  The  effective  sensitiveness  of  the 
method,  and  hence  the  accuracy  attainable  in  adjusting  the  con- 
tacts 2  and  3  for  the  condition  of  balance,  depends  on  the  above 


ABT.86]    BEST  MAGNITUDES  FOR  COMPONENTS       185 

factors  together  with  the  resistance  and  intrinsic  sensitiveness  of 
the  galvanometer. 

Since  RI  and  R%  are  adjusted  in  the  same  way  and  under  the 
same  conditions,  the  effective  sensitiveness  of  the  method  is  the 
same  for  both.  Consequently  only  one  of  them  will  be  considered 
in  the  present  discussion,  but  the  results  obtained  will  apply  with 
equal  rigor  to  either.  The  essential  parts  of  the  apparatus  and 
connections  are  illustrated  in  Fig.  12,  which  is  the  same  as  Fig.  10 
with  the  battery  B2  and  its  connections  omitted. 


FIG.  12. 

Let  V  =  e.m.f.  of  battery  BI, 

E  =  e.m.f.  of  battery  B3, 
R  =  resistance  between  1  and  2, 
W  =  total  resistance  of  the  circuit  1,  2,  Bs,  1, 
G  =  total  resistance  of  the  branch  1,  G,  BI,  2, 

I  =  current  through  B3) 

r  =  current  through  R, 

g  =  current  through  BI  and  G. 

When  the  contact  2  is  adjusted  to  the  balance  position 


Consequently 


=  0,    r  =  7,     and    7=^  =  -^ 


(xvii) 


This  is  the  fundamental  equation  of  the  potentiometer  and  must 
be  fulfilled  in  every  case  of  balance.     Consequently  E  must  be 


186  THE   THEORY   OF  MEASUREMENTS       [ART.  86 

chosen  larger  than  V  because  R  is  a  part  of  the  resistance  in  the 
circuit  1,  2,  Bz,  1,  and  hence  is  always  less  than  W.  Equation 
(xvii)  may  then  be  satisfied  by  a  suitable  adjustment  of  R. 

By  applying  Kirchhoff's  laws  to  the  circuits  1,  G,  BI,  2,  1,  and 
1,  2,  B3)  1,  when  the  contact  2  is  not  in  the  balance  position,  we 

have 

Rr-Gg=  V, 

and  Rr  +  (W  -  R)  I  =  E. 
But  r  =  I  -  g. 

Hence  RI-(R  +  G)g  =  V, 
and  WI  -  Rg  =  E. 

Eliminating  I  and  solving  for  g  we  find 

WV  -RE 


If  D  is  the  galvanometer  deflection  corresponding  to  the  current 
g  and  K  is  the  constant  of  the  instrument 

g  =  KD. 

Most  galvanometers  are,  or  can  be,  provided  with  interchange- 
able coils.  The  winding  space  in  such  coils  is  usually  constant, 
but  the  number  of  windings,  and  hence  the  resistance,  is  variable. 
Under  these  conditions  the  resistance  of  the  galvanometer  will  be 
approximately  proportional  to  the  square  of  the  number  of  turns 
of  wire  in  the  coils  used.  For  the  purpose  of  the  present  discussion, 
this  resistance  may  be  assumed  to  be  equal  to  G  since  the  resist- 
ance of  the  battery  and  connecting  wires  in  branch  1,  G,  BI,  2, 
can  usually  be  made  very  small  in  comparison  with  that  of  the 
galvanometer.  The  constant  K  is  inversely  proportional  to  the 
number  of  windings  in  the  coils  used.  Consequently,  as  a  suffi- 
ciently close  approximation  for  our  present  purpose,  we  have 

T 

v 

K  =  —  T=> 

VG 

where  T  is  a  constant  determined  by  the  dimensions  of  the  coils, 
the  moment  of  the  directive  couple,  and  various  other  factors 
depending  on  the  type  of  galvanometer  adopted.  Hence,  for  any 
given  instrument, 


ART.  86]     BEST  MAGNITUDES  FOR  COMPONENTS       187 

VG 

The  quantity  -jr  is  the  intrinsic  sensitiveness  of  the  galvanometer. 

It  is  equal  to  the  deflection  that  would  be  produced  by  unit  current 
if  the  instrument  followed  the  same  law  for  all  values  of  g. 
By  equation  (xix)  and  (xviii) 

VG        WV-RE 

T  *R*-WR-WG' 

The  variation  in  D  due  to  a  change  dR  in  R  is 


dD  VG  E(R*-WR-WG)  +  (WV-RE)(2R-W) 

dR  '   T  '  (R*-WR-WGY 

When  the  potentiometer  is  adjusted  for  a  balance,  D  is  equal  to 
zero  and  WV  is  equal  to  RE  by  equation  (xvii).  Hence,  if  d  is  the 
galvanometer  deflection  produced  when  the  resistance  R  is  changed 
from  the  balancing  value  by  an  amount  dR,  equation  (xx)  may 
be  put  in  the  form 

1  VVG 


The  fractional  change  in  R  corresponding  to  the  total  change  dR 
is 

.    I  '-f  •     :  I 

Consequently 

1          VVO 

~'  ' 


is  the  galvanometer  deflection  corresponding  to  a  fractional  error 
Pr  in  the  adjustment  of  R  for  balance.  The  coefficient  of  Pr  in 
equation  (xxi)  is  the  effective  sensitiveness  of  the  method  under 
the  given  conditions.  If  this  quantity  is  represented  by  S,  equa- 
tion (xxi)  becomes 

8  =  SPr, 

8      I 

° 


All  of  the  quantities  appearing  in  the  right-hand  member  of  this 
equation  may  be  considered  as  independent  variables  since  equa- 
tion (xvii)  can  always  be  satisfied,  and  hence  the  potentiometer 


188  THE  THEORY  OF  MEASUREMENTS       [ART.  86 

can  be  balanced,  when  R,  V,  and  E  have  any  assigned  values,  if 
the  resistance  W  is  suitably  chosen. 

If  d'  is  the  smallest  galvanometer  deflection  that  can  be  defi- 
nitely recognized  with  the  available  means  of  observation,  the  frac- 
tional error  P/  of  a  single  observation  on  R  should  not  be  greater 

5' 
than  -~  •     Since  the  precision  attainable  in  adj  usting  the  potentiom- 

o 

eter  for  balance  is  inversely  proportional  to  P/,  it  is  directly  pro- 
portional to  the  effective  sensitiveness  S.  By  choosing  suitable 
magnitudes  for  the  variables  T,  G,  R,  and  E,  it  is  usually  possible 
to  adjust  the  value  of  S,  and  hence  also  of  P/,  to  meet  the  re- 
quirements of  any  problem. 

From  equation  (xxii)  it  is  evident  that  S  will  increase  in  magni- 
tude continuously  as  the  quantities  T,  R,  and  E  decrease  and  that 
it  does  not  pass  through  a  maximum  value.  The  practicable  in- 
crease in  S  is  limited  by  the  following  considerations:  E  must  be 
greater  than  V,  for  the  reason  pointed  out  above,  and  its  variation 
is  limited  by  the  nature  of  available  batteries.  Since  E  must 
remain  constant  while  the  potentiometer  is  being  balanced  alter- 
nately against  V  and  the  electromotive  force  of  a  standard  cell, 
as  explained  in  article  eighty-one,  the  battery  B3  must  be  capable 
of  generating  a  constant  electromotive  force  during  a  considerable 
period  of  time.  In  practice  storage  cells  are  commonly  used  for 
this  purpose  and  E  may  be  varied  by  steps  of  about  two  volts  by 
connecting  the  required  number  of  cells  in  series.  Obviously  E 
should  be  made  as  nearly  equal  to  V  as  local  conditions  permit. 

When  the  potentiometer  is  balanced 

V      E 


If  R  is  reduced  for  the  purpose  of  increasing  the  effective  sensitive- 
ness, W  must  also  be  reduced  in  like  ratio,  and,  consequently,  the 
current  7  through  the  instrument  will  be  increased.  The  prac- 
tical limit  to  this  adjustment  is  reached  when  the  heating  effect 
of  the  current  becomes  sufficient  to  cause  an  appreciable  change 
in  the  resistances  R  and  W.  With  ordinary  resistance  boxes  this 
limit  is  reached  when  7  is  equal  to  a  few  thousandths  of  an  ampere. 
Consequently,  if  E  is  about  two  volts,  R  should  not  be  made  much 
less  than  one  thousand  ohms.  Resistance  coils  made  expressly 
for  use  in  a  potentiometer  can  be  designed  to  carry  a  much  larger 


ART.  86]    BEST  MAGNITUDES  FOR  COMPONENTS        189 

current  so  that  R  may  be  made  less  than  one  hundred  ohms  with- 
out introducing  serious  errors  due  to  the  heating  effect  of  the 
current. 

The  constant  T  depends  on  the  type  and  design  of  the  galva- 
nometer. In  the  suspended  magnet  type  it  can  be  varied  some- 
what by  changing  the  strength  of  the  external  magnetic  field,  and 
in  the  D'Arsonval  type  the  same  result  may  be  attained  by  chang- 
ing the  suspending  wires  of  the  movable  coil.  The  effects  of  the 
vibrations  of  the  building  in  which  the  instrument  is  located  and 
of  accidental  changes  in  the  external  magnetic  field  become  much 
more  troublesome  as  T  is  decreased,  i.e.,  as  the  intrinsic  sensitive- 
ness is  increased.  Consequently  the  practical  limit  to  the  reduc- 
tion of  T  is  reached  when  the  above  effects  become  sufficient  to 
render  the  observation  of  small  values  of  6  uncertain.  This  limit 
will  depend  largely  on  the  location  of  the  instrument  and  the  care 
that  is  taken  in  mounting  it.  Sometimes  a  considerable  reduc- 
tion in  T  can  be  effected  by  selecting  a  type  of  galvanometer 
suited  to  the  local  conditions. 

If  the  quantities  T7,  R,  V,  and  E  are  kept  constant,  S  passes 
through  a  maximum  value  when  G  satisfies  the  condition 


*?' 

It  can  be  easily  proved  by  direct  differentiation  that  this  is  the 
case  when 

G  = 


Hence,  after  suitable  values  of  the  other  variables  have  been  de- 
termined as  outlined  above,  the  best  magnitude  for  G  is  given  by 
equation  (xxiii).  Generally  this  condition  cannot  be  exactly  ful- 
filled in  practice  unless  a  galvanometer  coil  is  specially  wound  for 
the  purpose;  but,  when  several  interchangeable  coils  are  available, 
the  one  should  be  chosen  that  most  nearly  fulfills  the  condition. 
In  some  galvanometers  T  and  G  cannot  be  varied  independently, 
and  in  such  cases  suitable  values  can  be  determined  only  by  trial. 
Since  the  ease  and  rapidity  with  which  the  observations  can  be 
made  increase  with  T,  it  is  usually  advisable  to  adjust  the  other 
variables  to  give  the  greatest  practicable  value  to  the  second 
factor  in  S,  and  then  adjust  T  so  that  the  effective  sensitiveness 


190  THE   THEORY  OF  MEASUREMENTS       [ART.  86 

will  be  just  sufficient  to  give  the  required  precision  in  the  deter- 
mination of  R. 

As  an  illustration  consider  the  numerical  data  given  in  article 
eighty-one.  It  was  proved  that  the  specified  precision  require- 
ments cannot  be  satisfied  unless  R  is  determined  within  a  frac- 
tional precision  measure  equal  to  ±  0.00063.  Allowing  one-half 
of  this  to  errors  of  calibration  we  have  left  for  the  allowable  error 
in  adjusting  the  potentiometer 

Pr'  =  0.00031. 

If  a  single  storage  cell  is  used  at  B$,  E  is  approximately  two  volts, 
and,  with  ordinary  resistance  boxes,  R  should  be  about  one  thou- 
sand ohms,  for  the  reason  pointed  out  above.  This  condition  is 
fulfilled  by  the  cited  data;  and,  for  our  present  purpose,  it  will  be 
sufficiently  exact  to  take  V  equal  to  one  volt.  Hence,  by  equa- 
tion (xxiii),  the  most  advantageous  magnitude  for  G  is  about 
five  hundred  ohms;  and,  by  equation  (xxii),  the  largest  practi- 
cable value  for  the  second  factor  in  S  is 

ST  =          V  Jf =  0.0224. 

gf  1-41+0 


With  a  mirror  galvanometer  of  the  D'Arsonval  type,  read  by 
telescope  and  scale,  a  deflection  of  one-half  a  millimeter  can  be 
easily  detected.  Consequently,  if  we  express  the  galvanometer 
constant  K  in  terms  of  amperes  per  centimeter  deflection,  we  must 
take  5'  equal  to  0.05  centimeter;  and,  in  order  to  fulfill  the  specified 
precision  requirements,  the  effective  sensitiveness  must  satisfy  the 
condition 

S'          0.05 
~P7~00003l~ 

Combining  this  result  with  the  above  maximum  value  of  ST  we 
find  that  the  intrinsic  sensitiveness  must  be  such  that 

0.0224  _ 
161 

Hence  the  galvanometer  should  be  so  constructed  and  adjusted 
that 

G  =  500  ohms, 
and 

T 
K  =  •— =  =  6.2  X  lO"6  amperes  per  centimeter  deflection. 


ART.  86]    BEST  MAGNITUDES  FOR  COMPONENTS        191 

D'Arsonval  galvanometers  that  satisfy  the  above  specifications 
can  be  very  easily  obtained  and  are  much  less  expensive  than 
more  sensitive  instruments.  They  are  so  nearly  dead-beat  and 
free  from  the  effects  of  vibration  that  the  adjustment  of  the  poten- 
tiometer for  balance  can  be  easily  and  rapidly  carried  out  with 
the  necessary  precision.  Hence  the  use  of  such  an  instrument 
reduces  the  expense  of  the  measurements  without  increasing  the 
errors  of  observation  beyond  the  specified  limit. 


CHAPTER  XIII. 
RESEARCH. 

87.  Fundamental  Principles.  —  The  word  research,  as  used 
by  men  of  science,  signifies  a  detailed  study  of  some  natural 
phenomenon  for  the  purpose  of  determining  the  relation  between 
the  variables  involved  or  a  comparative  study  of  different  phe- 
nomena for  the  purpose  of  classification.  The  mere  execution  of 
measurements,  however  precise  they  may  be,  is  not  research.  On 
the  other  hand,  the  development  of  suitable  methods  of  measure- 
ment and  instruments  for  any  specific  purpose,  the  estimation  of 
unavoidable  errors,  and  the  determination  of  the  attainable  limit 
of  precision  frequently  demand  rigorous  and  far-reaching  research. 
As  an  illustration,  it  is  sufficient  to  cite  Michelson's  determination 
of  the  length  of  the  meter  in  terms  of  the  wave  length  of  light.  A 
repetition  of  this  measurement  by  exactly  the  same  method  and 
with  the  same  instruments  would  involve  no  research,  but  the 
original  development  of  the  method  and  apparatus  was  the  result 
of  careful  researches  extending  over  many  years. 

The  first  and  most  essential  prerequisite  for  research  in  any  field 
is  an  idea.  The  importance  of  research,  as  a  factor  in  the  advance- 
ment of  science,  is  directly  proportional  to  the  fecundity  of  the 
underlying  ideas. 

A  detailed  discussion  of  the  nature  of  ideas  and  of  the  conditions 
necessary  for  their  occurrence  and  development  would  lead  us  too 
far  into  the  field  of  psychology.  They  arise  more  or  less  vividly 
in  the  mind  in  response  to  various  and  often  apparently  trivial 
circumstances.  Their  inception  is  sometimes  due  to  a  flash  of 
intuition  during  a  period  of  repose  when  the  mind  is  free  to  respond 
to  feeble  stimuli  from  the  subconscious.  Their  development  and 
execution  generally  demand  vigorous  and  sustained  mental  effort. 
Probably  they  arise  most  frequently  in  response  to  suggestion  or 
as  the  result  of  careful,  though  tentative,  observations. 

A  large  majority  of  our  ideas  have  been  received,  in  more 
or  less  fully  developed  form,  through  the  spoken  or  written  dis- 
course of  their  authors  or  expositors.  Such  ideas  are  the  common 

192 


ART.  88]  RESEARCH  193 

heritage  of  mankind,  and  it  is  one  of  the  functions  of  research  to 
correct  and  amplify  them.  On  the  other  hand,  original  ideas, 
that  may  serve  as  a  basis  for  effective  research,  frequently  arise 
from  suggestions  received  during  the  study  of  generally  accepted 
notions  or  during  the  progress  of  other  and  sometimes  quite  differ- 
ent investigations. 

The  originality  and  productiveness  of  our  ideas  are  determined 
by  our  previous  mental  training,  by  our  habits  of  thought  and 
action,  and  by  inherited  tendencies.  Without  these  attributes, 
an  idea  has  very  little  influence  on  the  advancement  of  science. 
Important  researches  may  be,  and  sometimes  are,  carried  out  by 
investigators  who  did  not  originate  the  underlying  ideas.  But, 
however  these  ideas  may  have  originated,  they  must  be  so  thor- 
oughly assimilated  by  the  investigator  that  they  supply  the  stim- 
ulus and  driving  power  necessary  to  overcome  the  obstacles  that 
inevitably  arise  during  the  prosecution  of  the  work.  The  driving 
power  of  an  idea  is  due  to  the  mental  state  that  it  produces  in  the 
investigator  whereby  he  is  unable  to  rest  content  until  the  idea 
has  been  thoroughly  tested  in  all  its  bearings  and  definitely  proved 
to  be  true  or  false.  It  acts  by  sustaining  an  effective  concentra- 
tion of  the  mental  and  physical  faculties  that  quickens  his  in- 
genuity, broadens  his  insight,  and  increases  his  dexterity. 

In  order  to  become  effective,  an  idea  must  furnish  the  incentive 
for  research,  direct  the  development  of  suitable  methods  of  pro- 
cedure, and  guide  the  interpretation  of  results.  But  it  must 
never  be  dogmatically  applied  to  warp  the  facts  of  observation 
into  conformity  with  itself.  The  mind  of  the  investigator  must 
be  as  ready  to  receive  and  give  due  weight  to  evidence  against 
his  ideas  as  to  that  in  their  favor.  The  ultimate  truth  regarding 
phenomena  and  their  relations  should  be  sought  regardless  of 
the  collapse  of  generally  accepted  or  preconceived  notions.  From 
this  point  of  view,  research  is  the  process  by  which  ideas  are 
tested  in  regard  to  their  validity. 

88.  General  Methods  of  Physical  Research.  —  Researches 
that  pertain  to  the  physical  sciences  may  be  roughly  classified 
in  two  groups:  one  comprising  determinations  of  the  so-called 
physical  constants  such  as  the  atomic  weights  of  the  elements,  the 
velocity  of  light,  the  constant  of  gravitation,  etc.;  the  other 
containing  investigations  of  physical  relations  such  as  that  which 
connects  the  mass,  volume,  .pressure,  and  temperature  of  a  gas. 


194  THE   THEORY  OF  MEASUREMENTS       [ART.  88 

The  researches  in  the  first  group  ultimately  reduce  to  a  careful 
execution  of  direct  or  indirect  measurements  and  a  determination 
of  the  precision  of  the  results  obtained.  The  general  principles 
that  should  be  followed  in  this  part  of  the  work  have  been  suffi- 
ciently discussed  in  preceding  chapters.  Their  application  to  prac- 
tical problems  must  be  left  to  the  ingenuity  and  insight  of  the 
investigator.  Some  men,  with  large  experience,  make  such  appli- 
cations almost  intuitively.  But  most  of  us  must  depend  on  a 
more  or  less  detailed  study  of  the  relative  capabilities  of  available 
methods  to  guide  us  in  the  prosecution  of  investigations  and  in 
the  discussion  of  results. 

In  general,  physical  constants  do  not  maintain  exactly  the  same 
numerical  value  under  all  circumstances,  but  vary  somewhat  with 
changes  in  surrounding  conditions  or  with  lapse  of  time.  Thus 
the  velocity  of  light  is  different  in  different  media  and  in  dispersive 
media  it  is  a  function  of  the  frequency  of  the  vibrations  on  which 
it  depends.  Consequently  the  determination  of  such  constants 
should  be  accompanied  by  a  thorough  study  of  all  of  the  factors 
that  are  likely  to  affect  the  values  obtained  and  an  exact  specifica- 
tion of  the  conditions  under  which  the  measurements  are  made. 
Such  a  study  frequently  involves  extensive  investigations  of  the 
phenomena  on  which  the  constants  depend  and  it  should  be 
carried  out  by  very  much  the  same  methods  that  apply  to  the 
determination  of  physical  relations  in  general.  On  the  other 
hand,  the  exact  expression  of  a  physical  relation  generally  involves 
one  or  more  constants  that  must  be  determined  by  direct  or  in- 
direct measurements.  Hence  there  is  no  sharp  line  of  division 
between  the  first  and  second  groups  specified  above,  many  re- 
searches belonging  partly  to  one  group  and  partly  to  the  other. 

The  occurrence  of  any  phenomenon  is  usually  the  result  of  the 
coexistence  of  a  number  of  more  or  less  independent  antecedents. 
Its  complete  investigation  requires  an  exact  determination  of  the 
relative  effect  of  each  of  the  contributary  causes  and  the  develop- 
ment of  the  general  relation  by  which  their  interaction  is  expressed. 
A  determination  of  the  nature  and  mode  of  action  of  all  of  the 
antecedents  is  the  first  step  in  this  process.  Since  it  is  gen- 
erally impossible  to  derive  useful  information  by  observing  the 
combined  action  of  a  number  of  different  causal  factors,  it  becomes 
necessary  to  devise  means  by  which  the  effects  of  the  several 
factors  can  be  controlled  in  such  manner  that  they  can  be  studied 


ART.  88]  RESEARCH  195 

separately.  The  success  of  researches  of  this  type  depends  very 
largely  on  the  effectiveness  of  such  means  of  control  and  the 
accuracy  with  which  departures  from  specified  conditions  can  be 
determined. 

Suppose  that  an  idea  has  occurred  to  us  that  a  certain  phenome- 
non is  due  to  the  interaction  of  a  number  of  different  factors  that 
we  will  represent  by  A,  B,  C,  .  .  .  ,  P.  This  idea  may  involve 
a  more  or  less  definite  notion  regarding  the  relative  effects  of  the 
several  factors  or  it  may  comprehend  only  a  notion  that  they  are 
connected  by  some  functional  relation.  In  either  case  we  wish 
to  submit  our  idea  to  the  test  of  careful  research  and  to  determine 
the  exact  form  of  the  functional  relation  if  it  exists. 

The  investigation  is  initiated  by  making  a  series  of  preliminary 
observations  of  the  phenomenon  corresponding  to  as  many  vari- 
ations in  the  values  of  the  several  factors  as  can  be  easily  effected. 
The  nature  of  such  observations  and  the  precision  with  which  they 
should  be  made  depend  so  much  on  the  character  of  the  problem 
in  hand  that  it  would  be  impossible  to  give  a  useful  general  dis- 
cussion of  suitable  methods  of  procedure.  Sometimes  roughly 
quantitative,  or  even  qualitative,  observations  are  sufficient.  In 
other  cases  a  considerable  degree  of  precision  is  necessary  before 
definite  information  can  be  obtained.  In  all  cases  the  observa- 
tions should  be  sufficiently  extensive  and  exact  to  reveal  the  gen- 
eral nature  and  approximate  relative  magnitudes  of  the  effects 
produced  by  each  of  the  factors.  They  should  also  serve  to  detect 
the  presence  of  factors  not  initially  contemplated. 

With  the  aid  of  the  information  derived  from  preliminary  obser- 
vations and  from  a  study  of  such  theoretical  considerations  as 
they  may  suggest,  means  are  devised  for  exactly  controlling  the 
magnitude  of  each  of  the  factors.  Methods  are  then  developed 
for  the  precise  measurement  of  these  magnitudes  under  the  con- 
ditions imposed  by  the  adopted  means  of  control.  This  process 
often  involves  a  preliminary  trial  of  several  different  methods 
for  the  purpose  of  determining  their  relative  availability  and  pre- 
cision. The  methods  that  are  found  to  be  most  exact  and  con- 
venient usually  require  some  modification  to  adapt  them  to  the 
requirements  of  a  particular  problem.  Sometimes  it  becomes 
necessary  to  devise  and  test  entirely  new  methods.  During  this 
part  of  the  investigation  the  discussions  of  the  precision  of  meas- 
urements given  in  the  preceding  chapters  find  constant  applica- 


196  THE  THEORY  OF  MEASUREMENTS       [ART.  88 

tion  and  it  is  largely  through  them  that  the  suitableness  of 
proposed  methods  is  determined. 

After  definite  methods  of  measurement  and  means  of  control 
have  been  adopted  and  perfected  to  the  required  degree  of  pre- 
cision, the  final  measurements  on  the  factors,  A,  B,  C,  .  .  .  ,  P, 
are  carried  out  under  the  conditions  that  are  found  to  be  most 
advantageous.  Usually  two  of  the  factors,  say  A  and  B,  are 
caused  to  vary  through  as  large  a  range  of  values  as  conditions 
will  permit  while  the  other  factors  are  maintained  constant  at 
definite  observed  values.  At  stated  intervals  the  progress  of  the 
variation  is  arrested  and  corresponding  values  of  A  and  B  are 
measured  while  they  are  kept  constant.  From  a  sufficiently 
extended  series  of  such  observations  it  is  usually  possible  to  make 
an  empirical  determination  of  the  form  of  the  functional  relation 

A  =/i(£);     C,Z>,  .  .  .  ,P.  constant.  (i) 

On  the  other  hand,  if  the  form  of  the  function  /i  is  given  as  a 
theoretical  deduction  from  the  idea  underlying  the  investigation, 
the  observations  serve  to  test  the  exactness  of  the  idea  and  de- 
termine the  magnitudes  of  the  constants  involved  in  the  given 
function.  By  allowing  different  factors  to  vary  and  making 
corresponding  measurements,  the  relations 

A  =/2(C);     B,D,  .  .      ,  P,  constant, 


A  =/n(P);    £,C,Z>,  •  •  .,  constant, 


(ii) 


may  be  empirically  determined  or  verified.  As  many  functions  of 
this  type  as  there  are  pairs  of  factors  might  be  determined,  but 
usually  it  is  not  necessary  to  establish  more  than  one  relation  for 
each  factor.  Generally  it  is  convenient  to  determine  one  of  the 
factors  as  a  function  of  each  of  the  others  as  illustrated  above; 
but  it  is  not  necessary  to  do  so,  and  sometimes  the  determination 
of  a  different  set  of  relations  facilitates  the  investigation. 

During  the  establishment  of  the  relation  between  any  two 
factors  all  of  the  others  are  supposed  to  remain  rigorously  con- 
stant. Frequently  this  condition  cannot  be  exactly  fulfilled  with 
available  means  of  control,  but  the  variations  thus  introduced 
can  usually  be  made  so  small  that  their  effects  can  be  treated  as 
constant  errors  and  removed  with  the  aid  of  the  relations  after- 
wards found  to  exist  between  the  factors  concerned,  For  this 


ART.  88]  RESEARCH  197 

purpose  frequent  observations  must  be  made  on  the  factors  that 
are  supposed  to  remain  constant  during  the  measurement  of  the 
two  principal  variables.  If  the  variations  in  these  factors  are  not 
very  small  all  of  the  relations  determined  by  the  principal  measure- 
ments will  be  more  or  less  in  error  and  must  be  treated  as  first 
approximations.  Usually  such  errors  can  be  eliminated  and  the 
true  relations  established  with  sufficient  precision,  by  a  series  of 
successive  approximations.  However,  the  weight  of  the  final 
result  increases  very  rapidly  with  the  effectiveness  of  the  means  of 
control  and  it  is  always  worth  while  to  exercise  the  care  necessary 
to  make  them  adequate. 

When  the  functions  involved  in  equations  (i)  and  (ii),  or  their 
equivalents  in  terms  of  other  combinations  of  factors,  have  been 
determined  with  sufficient  precision,  they  can  usually  be  com- 
bined into  a  single  relation,  in  the  form 


or 


A=F(B,C,D,  .  .  .  ,P), 
F(A,B,C,D,  ,P)=0, 


(iii) 


which  expresses  the  general  course  of  the  investigated  phenomenon 
in  response  to  variations  of  the  factors  within  the  limits  of  the 
observations.  Such  generalizations  may  be  purely  empirical  or 
they  may  rest  partly  or  entirely  on  theoretical  deductions  from 
well-established  principles.  In  either  case  the  test  of  their  validity 
lies  in  the  exactness  with  which  they  represent  observed  facts. 
While  an  exact  empirical  formula  finds  many  useful  applications 
in  practical  problems  it  should  not  be  assumed  to  express  the  true 
physical  nature  of  the  phenomenon  it  represents.  In  fact  our 
understanding  of  any  phenomenon  is  but  scanty  until  we  can 
represent  its  course  by  a  formula  that  gives  explicit  or  implicit 
expression  to  the  physical  principles  that  underlie  it.  Conse- 
quently a  research  ought  not  to  be  considered  complete  until  the 
investigated  phenomenon  has  been  classified  and  represented  by  a 
function  that  exhibits  the  physical  relations  among  its  factors. 
(i  It  is  scat cely  necessary  to  point  out  that  a  complete  research 
as  outlined  above  is  seldom  carried  out  by  one  man  and  that  the 
underlying  ideas  very  rarely  originate  at  the  same  time  or  in  the 
same  person.  The  preliminary  relations  in  the  form  of  equations 
(i)  and  (ii)  are  frequently  inspired  by  independent  ideas  and 
worked  out  by  different  men.  The  exact  determination  of  any 


198  THE  THEORY  OF  MEASUREMENTS       [ART.  89 

one  of  them  constitutes  a  research  that  is  complete  so  far  as  it 
goes.  The  establishment  of  the  general  relation  that  compre- 
hends all  of  the  others  and  the  interpretation  of  its  physical  signifi- 
cance are  generally  the  result  of  a  process  of  gradual  growth  and 
modification  to  which  many  men  have  contributed. 

89.  Graphical  Methods  of  Reduction.  —  After  the  necessary 
measurements  have  been  completed  and  corrected  for  all  known 
constant  errors,  the  form  of  the  functions  appearing  in  equations 
(i)  and  (ii),  or  other  equations  of  similar  type,  and  the  numerical 
value  of  the  constants  involved  can  sometimes  be  determined 
easily  and  effectively  by  graphical  methods.  Such  methods  are 
almost  universally  adopted  for  the  discussion  of  preliminary  obser- 
vations and  the  determination  of  approximate  values  of  the  con- 
stants. In  some  cases  they  are  the  only  methods  by  which  the 
results  of  the  measurements  can  be  expressed.  In  some  other 
cases  the  constants  can  be  more  exactly  determined  by  an  appli- 
cation of  the  method  of  least  squares  to  be  described  later.  Usu- 
ally, however,  the  general  form  of  the  functions  and  approximate 
values  of  the  constants  must  first  be  determined  by  graphical 
methods  or  otherwise. 

Let  x  and  y  represent  the  simultaneous  values  of  two  variable 
factors  corresponding  to  specified  constant  values  of  the  other 
factors  involved  in  the  phenomenon  under  investigation.  Suppose 
that  x  has  been  varied  by  successive  nearly  equal  steps  through 
as  great  a  range  as  conditions  permit  and  that  the  simultaneous 
values  x  and  y  have  been  measured  after  each  of  these  steps  while 
the  factors  that  they  represent  were  kept  constant.  If  all  other 
factors  have  remained  constant  throughout  these  operations,  the 
above  series  of  measurements  on  x  and  y  may  be  applied  at  once 
to  the  determination  of  the  form  and  constants  of  the  functional 
relation 


This  expression  is  of  the  same  type  as  equations  (i.)  and  (ii). 
Consequently  the  following  discussion  applies  generally  to  all 
cases  in  which  there  are  only  two  variable  factors.  If  the  sup- 
posedly constant  factors  are  not  strictly  constant  during  the 
measurements,  the  observations  on  x  and  y  will  not  give  the  true 
form  of  the  function  in  (iv)  until  they  have  been  corrected  for 
the  effects  of  the  variations  thus  introduced. 


ART.  89]  RESEARCH  199 

As  the  first  step  in  the  graphical  method  of  reduction,  the 
observations  on  x  and  y  are  laid  off  as  abscissae  and  ordinates  on 
accurately  squared  paper,  and  the  points  determined  by  corre- 
sponding coordinates  are  accurately  located  with  a  fine  pointed 
needle.  The  visibility  of  these  points  is  usually  increased  by 
drawing  a  small  circle  or  other  figure  with  its  center  exactly  at 
the  indicated  point.  The  scale  of  the  plot  should  be  so  chosen 
that  the  form  of  the  curve  determined  by  the  located  points  is 
easily  recognized  by  eye.  In  order  to  bring  out  the  desired  rela- 
tion, it  is  frequently  necessary  to  adopt  a  different  scale  for  ordi- 
nates and  abscissae.  Usually  it  is  advantageous  to  choose  such 
scales  that  the  total  variations  of  x  and  y  will  be  represented  by 
approximately  equal  spaces.  Thus,  if  the  total  variation  of  y  is 
numerically  equal  to  about  one-tenth  of  the  corresponding  vari- 
ation of  x,  the  i/'s  should  be  plotted  to  a  scale  approximately  ten 
times  as  large  as  that  adopted  for  the  x's.  In  all  cases  the  adopted 
scales  should  be  clearly  indicated  by  suitable  numbers  placed  at 
equal  intervals  along  the  vertical  and  horizontal  axes.  Letters 
or  other  abbreviations  should  be  placed  near  the  ends  of  the  axes 
to  indicate  the  quantities  represented. 

The  points  thus  located  usually  lie  very  nearly  on  a  uniform 
curve  that  represents  the  functional  relation  (iv).  Consequently 
the  problem  in  hand  may  be  solved  by  determining  the  equation 
of  this  curve  and  the  numerical  value  of  the  constants  involved 
in  it.  Sometimes  it  is  impossible  or  inadvisable  to  carry  out  such 
a  determination  in  practice  and  in  such  cases  the  plotted  curve 
is  the  only  available  means  of  representing  the  relation  between 
the  observed  factors.  In  all  cases  the  deviations  of  the  located 
points  from  the  uniform  curve  represent  the  residuals  of  the 
observations,  and,  consequently,  indicate  the  precision  of  the 
measurements  on  x  and  y. 

The  simplest  case,  and  one  that  frequently  occurs  in  practice,  is 
illustrated  in  Fig.  13.  The  plotted  points  lie  very  nearly  on  a 
straight  line.  Consequently  the  functional  relation  (iv)  takes  the 
linear  form 

y  =  Ax  +  B,  (v) 

where  A  is  the  tangent  of  the  angle  a  between  the  line  and  the 
positive  direction  of  the  x  axis,  and  B  is  the  intercept  OP  on 
the  y  axis.  For  the  determination  of  the  numerical  values  of  the 


200 


THE   THEORY  OF  MEASUREMENTS        [ART.  89 


constants  A  and  B,  the  line  should  be  sharply  drawn  in  such  a 
position  that  the  plotted  points  deviate  from  it  about  equally  in 
opposite  directions,  i.e.,  the  sum  of  the  positive  deviations  should 
be  made  as  nearly  as  possible  equal  to  the  sum  of  the  negative 
deviations.  If  this  has  been  carefully  and  accurately  done,  the 
constant  B  may  be  determined  by  a  direct  measurement  of  the 
intercept  OP  in  terms  of  the  scale  used  in  plotting  the  y's- 


0.10 


05 


25 


FIG.  13. 


50 


75 


The  constant  A  may  be  computed  from  measurements  of  the 
coordinates  x\  and  2/1  of  any  point  on  the  line,  not  one  of  the  plotted 
points,  by  the  relation 


If  the  position  of  the  line  is  such  that  the  point  P  does  not  fall 
within  the  limits  of  the  plotting  sheet,  the  coordinates,  Xi,  y\  and 
£2,  2/2,  of  two  points  on  the  line  are  measured.  Since  they  must 
satisfy  equation  (v), 

2/i  =  Axi  +  B, 
and 

2/2  =  Ax2  +  B. 
Consequently 

A  =  — and    B  — 


—  X2 


The  points  selected  for  this  purpose  should  be  as  widely  separated 
as  possible  in  order  to  reduce  the  effect  of  errors  of  plotting  and 


ART.  89]  RESEARCH  201 

measurement.  The  accuracy  of  these  determinations  is  likely  to 
be  greatest  when  the  vertical  and  horizontal  scales  are  so  chosen 
that  the  line  makes  an  angle  of  approximately  forty-five  degrees 
with  the  x  axis.  Space  may  sometimes  be  saved  and  the  appear- 
ance of  the  plot  improved  by  subtracting  a  constant  quantity, 
nearly  equal  to  B,  from  each  of  the  y's  before  they  are  plotted. 

Many  physical  relations  are  not  linear  in  form.  Perhaps  none 
of  them  are  strictly  linear  when  large  ranges  of  variation  are  con- 
sidered. Consequently  the  plotted  points  are  more  likely  to  lie 
nearly  on  some  regular  curve  than  on  a  straight  line.  In  such 
cases  the  form  of  the  functional  relation  (iv)  is  sometimes  sug- 
gested by  theoretical  considerations,  but  frequently  it  must  be 
determined  by  the  method  of  trial  and  error  or  successive  approxi- 
mations. For  this  purpose  the  curve  representing  the  observa- 
tions is  compared  with  a  number  of  curves  representing  known 
equations.  The  equation  of  the  curve  that  comes  nearest  to  the 
desired  form  is  modified  by  altering  the  numerical  values  of  its 
constants  until  it  represents  the  given  measurements  within  the 
accidental  errors  of  observation.  Frequently  several  different 
equations  and  a  number  of  modifications  of  the  constants  must 
be  tried  before  satisfactory  agreement  is  obtained. 

When  the  desired  relation  does  not  contain  more  than  two  inde- 
pendent constants,  it  can  sometimes  be  reduced  to  a  linear  relation 
between  simple  functions  of  x  and  y.  Thus,  the  equation 

y  =  Be~Ax,     .  (vi) 

represented  by  the  curve  in  Fig.  14,  is  frequently  met  with  in 
physical  investigations.  By  inverting  (vi)  and  introducing '  log- 
arithms, we  obtain  the  relation 

log*  y  =  log*  B  -  Ax. 

Hence  if  the  logarithms  of  the  y's  are  laid  off  as  ordinates  against 
the  corresponding  x's  as  abscissae,  the  located  points  will  lie  very 
nearly  on  a  straight  line  if  the  given  observations  satisfy  the  func- 
tional relation  (vi) .  When  this  is  the  case,  the  constants  A  and 
loge  B  may  be  determined  by  the  methods  developed  during  the 
discussion  of  equation  (v).  If  logarithms  to  the  base  ten  are 

used  the  above  equation  becomes 

^| 
log™  y  =  logio  B  -       x, 


202 


THE   THEORY  OF  MEASUREMENTS       [ART.  89 


where  M  is  the  modulus  of  the  natural  system  of  logarithms.     In 

^ 
this  case  the  plot  gives  the  values  of  logio  B  and  -^  from  which 

the  constants  A  and  B  can  be  easily  computed.     When  the  plotted 
points  do  not  lie  nearer  to  a  straight  line  than  to  any  other  curve, 
y 


10 


\ 


\ 


0.5 


1.0 


1.5 


FIG.  14. 


equation  (vi)  does  not  represent  the  functional  relation  between 
the  observed  factors  and  some  other  form  must  be  tried.  Many 
of  the  commonly  occurring  forms  may  be  treated  by  the  above 
method  and  the  process  is  usually  so  simple  that  further  illustra- 
tion seems  unnecessary. 

The  curve  determined  by  plotting  the  x's  and  y's  directly  fre- 
quently exhibits  points  of  discontinuity  or  sharp  bends  as  at  p 
and  q  in  Fig.  15.  Such  irregularities  are  generally  due  to  changes 
in  the  state  of  the  material  under  investigation.  The  nature-  and 
causes  of  such  changes  are  frequently  determined,  or  at  least 
suggested,  by  the  location  and  character  of  such  points.  The 
different  branches  of  the  curve  may  correspond  to  entirely  differ- 
ent equations  or  to  equations  in  the  same  form  but  with  different 
constants.  In  either  case  the  equation  of  each  branch  must  be 
determined  separately. 

The  accuracy  attainable  by  graphical  methods  depends  very 
largely  on  the  skill  of  the  draughtsman  in  choosing  suitable  scales 
and  executing  the  necessary  operations.  In  many  cases  the  errors 


ART.  90] 


RESEARCH 


203 


due  to  the  plot  are  less  than  the  errors  of  observation  and  it  would 
be  useless  to  adopt  a  more  precise  method  of  reduction.  When 
the  means  of  control  are  so  well  devised  and  effective  that  the 
constant  errors  left  in  the  measurements  are  less  than  the  errors 
of  plotting  it  is  probably  worth  while  to  make  the  reductions  by 
the  method  of  least  squares,  as  explained  in  the  following  article. 
y 


'FiG.  15. 

90.  Application  of  the  Method  of  Least  Squares.  —  In  the 

case  of  linear  relations,  expressible  in  the  form  of  equation  (v), 
the  best  values  of  the  constants  A  and  B  can  be  very  easily  deter- 
mined by  applying  the  method  of  least  squares  in  the  manner 
explained  in  article  fifty-one.  However,  as  pointed  out  in  the 
preceding  article,  very  few  physical  relations  are  strictly  linear 
when  large  variations  of  the  involved  factors  are  considered. 
Consequently  a  straight  line,  corresponding  to  constants  deter- 
mined as  above,  usually  represents  only  a  small  part  of  the  course 
of  the  investigated  phenomenon.  Such  a  line  is  generally  a  short 
chord  of  the  curve  that  represents  the  true  relation  and  conse- 
quently its  direction  depends  on  the  particular  range  covered  by 
the  observations  from  which  it  is  derived. 

When  the  measurements  are  extended  over  a  sufficiently  wide 
range,  the  points  plotted  from  them  usually  deviate  from  a  straight 
line  in  an  approximately  regular  manner,  as  illustrated  in  Fig.  16, 


204 


THE   THEORY  OF  MEASUREMENTS       [ART.  90 


and  lie  very  near  to  a  continuous  curve  of  slight  curvature.  Meas- 
urements of  this  type  can  always  be  represented  empirically  by  a 
power  series  in  the  form 

y  =  A  +  Bx  +  Cx*  +  .  -  -  ,  (vii) 

the  number  of  terms  and  the  signs  of  the  constants  depending  on 
the  magnitude  and  sign  of  the  curvature  to  be  represented. 


FIG.  16. 

Since  equation  (vii)  is  linear  with  respect  to  the  constants  A,  B, 
C,  etc.,  they  might  be  computed  directly  from  the  observations 
on  x  and  y  by  the  method  of  least  squares.  Usually,  however, 
the  computations  can  be  simplified  by  introducing  approximate 
values  of  the  constants  A  and  B.  Thus,  let  A'  and  B'  represent 
two  numerical  quantities  so  chosen  that  the  line 

y'  =  A'  +  B'x 

passes  in  the  same  general  direction  as  the  plotted  points,  in  the 
manner  illustrated  by  the  dotted  line  in  pig.  16.  The  difference 
between  y  and  y'  can  be  put  in  the  form 

y  —  y'  =  (A  —  A')  +  MI  (B  —  B'}  -^  4-  M2C  —  +  .  .  .       (viii) 

MI  M2 

where  Afi,  M2,  etc.,  represent  numerical  constants  so  chosen  that 

*Y*  s¥»2 

the  quantities  y  -  y',  -=—,  —   etc.,  are  nearly  of  the  same  order 


ART.  90]  RESEARCH  205 

of  magnitude.     For  the  sake  of  convenience  let 

(ix) 
and 


The  quantities  s,  6,  c,  etc.,  may  be  derived  from  the  observations, 
with  the  aid  of  the  assumed  constants  A',  B',  MI,  Mz,  etc.;  and  xi, 
xz,  xS}  etc.,  are  the  unknowns  to  be  computed  by  the  method  of 
least  squares.  After  the  above  substitutions,  equation  (viii)  takes 
the  simple  form 

xi  +  bx2  +  cx3  +  •  •  •   =  s, 

which  is  identical  with  that  of  the  observation  equations  (53), 
article  forty-nine.  As  many  equations  of  this  type  may  be  formed 
as  there  are  pairs  of  corresponding  measurements  on  x  andj  y. 

The  normal  equations  (56)  may  be  derived  from  the  observation 
equations  thus  established,  by  the  methods  explained  in  articles 
fifty  and  fifty-three.  Their  final  solution  for  the  unknowns  Xi,  Xz, 
xsj  etc.,  may  be  effected  by  Gauss's  method,  developed  in  article 
fifty-four  and  illustrated  in  article  fifty-five,  or  by  any  other  con- 
venient method.  The  corresponding  numerical  values  of  the 
constants  A,  B,  C,  etc.,  may  then  be  computed  by  equations  (ix). 
These  values,  when  substituted  in  (vii) ,  give  the  required  empirical 
relation  between  x  and  y. 

If  a  sufficient  number  of  terms  have  been  included  in  equation 
(vii),  the  relation  thus  established  will  represent  the  given  measure- 
ments within  the  accidental  errors  of  observation.  The  residuals, 
computed  by  equations  (54),  article  forty-nine,  and  arranged  in 
the  order  of  increasing  values  of  y,  should  show  approximately  as 
many  sign  changes  as  sign  follows.  When  this  is  not  the  case 
the  observed  y's  deviate  systematically  from  the  values  given  by 
equation  (vii)  for  corresponding  x's.  In  such  cases  the  number  of 
terms  employed  is  not  sufficient  for  the  exact  representation  of  the 
observed  phenomenon,  and  a  new  relation  in  the  same  general 
form  as  the  one  already  tested  but  containing  more  independent 
constants  should  be  determined.  This  process  must  be  repeated 
until  such  a  relation  is  established  that  systematically  varying 
differences  between  observed  and  computed  y's  no  longer  occur. 

The  observation  equations  used  as  a  basis  for  the  numerical 
illustration  given  in  article  fifty-five  were  derived  from  the  follow- 


206 


THE   THEORY  OF  MEASUREMENTS       [ART.  90 


ing  observations  on  the  thermal  expansion  of  petroleum  by  equa- 
tions (viii)  and  (ix),  taking 

A'  =  1000;    B'  =  l;    Ml  =  10;    and    M2  =  1000. 


X 

temperature 

volume 

degrees 

cc. 

0 

1000.24 

20 

1018.82 

40 

1038.47 

60 

1059  31 

80 

1081.20 

100 

1104.27 

The  computations  carried  out  in  the  cited  article  resulted  as 

follows : 

xi  =  0.245;        x2  =  -  1.0003;        x3  =  1.4022. 

Hence,  by  equations  (ix) 

A  =  1000.245;  B  =  0.89997;        C  =  0.0014022, 

and  the  functional  relation  (vii)  becomes 

y  =  1000.245  +  0.89997  •  x  +  0.0014022  •  x\ 

The  residuals  corresponding  to  this  relation,  computed  and  tab- 
ulated in  article  fifty-five,  show  five  sign  changes  and  no  sign 
follows.  Such  a  distribution  of  signs  sometimes  indicates  that  the 
observed  factors  deviate  periodically  from  the  assumed  functional 
relation.  In  the  present  case,  however,  the  number  of  observa- 
tions is  so  small  that  the  apparent  indications  of  the  residuals  are 
probably  fortuitous.  Consequently  it  would  not  be  worth  while 
to  repeat  the  computations  with  a  larger  number  of  terms  unless 
it  could  be  shown  by  independent  means  that  the  accidental  errors 
of  the  observations  are  less  than  the  residuals  corresponding  to  the 
above  relation. 

Any  continuous  relation  between  two  variables  can  usually  be 
represented  empirically  by  an  expression  in  the  form  of  equation 
(vii).  However,  it  frequently  happens  that  the  physical  signifi- 
cance of  the  investigated  phenomenon  is  not  suggested  by  such 
an  expression  but  is  represented  explicitly  by  a  function  that  is  not 
linear  with  respect  to  either  the  variable  factors  or  the  constants 
involved.  Such  functions  usually  contain  more  than  two  inde- 
pendent constants  and  sometimes  include  more  than  two  variable 
factors.  They  may  be  expressed  by  the  general  equation 

y  =  F(A,B,C,.         ,x,z,.      .  ),  (154) 


ART.  90]  RESEARCH  207 

where  A,  B,  C,  etc.,  represent  constants  to  be  determined  and  yt  x, 
z,  etc.,  represent  corresponding  values  of  observed  factors. 

Sometimes  the  form  of  the  function  F  is  given  by  theoretical 
considerations,  but  more  frequently  it  must  be  determined,  to- 
gether with  the  numerical  values  of  the  constants,  by  the  method 
of  successive  approximations.  In  the  latter  case  a  definite  form, 
suggested  by  the  graphical  representation  of  the  observations  or 
by  analogy  with  similar  phenomena,  is  assumed  tentatively  as  a 
first  approximation.  Then,  by  substituting  a  number  of  different 
corresponding  observations  on  y,  x,  z,  etc.,  in  (154),  as  many  inde- 
pendent equations  as  there  are  constants  in  the  assumed  function 
are  established.  The  simultaneous  solution  of  these  equations 
gives  first  approximations  to  the  values  of  the  constants  A,  B,  C, 
etc.  Sometimes  the  solution  cannot  be  effected  directly  by  means 
of  the  ordinary  algebraic  methods,  but  it  can  usually  be  accom- 
plished with  sufficient  accuracy  either  by  trial  and  error  or  by 
some  other  method  of  approximation. 

Let  A',  B'  ',  C',  etc.,  represent  approximate  values  of  the  con- 
stants and  let  61,  52,  53,  etc.,  represent  their  respective  deviations 
from  the  true  values.  Then 

A=A'  +  51;    B  =  B'  +  d2]    C  =  C'  +  53,  etc.,       (155) 
and  (154)  may  be  put  in  the  form 
y-F\(A'  +  Sd,     (B'  +  fc),     (C"  +  «.)  ----  ,*,*,  .  -  .  |    (x) 

If  the  S's  are  so  small  that  their  squares  and  higher  powers  may 
be  neglected,  expansion  by  Taylor's  Theorem  gives 


y-F(A',B',C',  .  .  .  ,x,z,  .  . 
dF  dF      ,      dF 


,,.  .  .,,,.. 

By  putting 

y-F(A',B',C',  .  .  .  ,x,z,  .  .  .  )  =  «; 

(156) 


and  transposing,  equation  (xi)  becomes 

adi  +  652  +  c53  +  .  .  .   =  s.  (157) 

As  many  independent  equations  of  this  type  as  there  are  sets  of 
corresponding  observations  on  y,  x,  z,  etc.,  can  be  formed.  The 
absolute  term  s  and  the  coefficients  a,  6,  c,  etc.,  in  each  equation 
are  computed  from  a  single  set  of  observations  by  the  relations 


208  THE  THEORY  OF  MEASUREMENTS       [ART.  90 

(156)  with  the  aid  of  the  approximate  values  A',  Bf,  C",  etc.  Since 
the  resulting  equations  are  in  the  same  form  as  the  observation 
equations  (53),  the  normal  equations  (56)  may  be  found  and 
solved  by  the  methods  described  in  Chapter  VII.  The  values 
of  $1,  62,  53,  etc.,  thus  obtained,  when  substituted  in  (155),  give 
second  approximations  to  the  values  of  the  constants  A,  B,  C, 
etc. 

The  accuracy  of  the  second  approximations  will  depend  on  the 
assumed  form  of  the  function  F  and  on  the  magnitude  of  the  correc- 
tions Si,  62,  63,  etc.  If  these  corrections  are  not  small,  the  con- 
ditions underlying  equation  (xi)  are  not  fulfilled  and  the  results 
obtained  by  the  above  process  may  deviate  widely  from  the  correct 
values  of  the  constants;  but,  except  in  extreme  cases,  they  are 
more  accurate  than  the  first  approximations  A',  Bf,  C',  etc.  Let 
A",  B",  C",  etc.,  represent  the  second  approximations.  The 
corresponding  residuals,  n,  r2,  .  .  .  ,  rn,  may  be  computed  by 
substituting  different  sets  of  corresponding  observations  on  y, 
x,  z,  etc.,  successively  in  the  equation 

F(A",B",C",  .  .  .  ,x,z,  .  .  .  )-y  =  r,  (xii) 

where  the  function  F  has  the  same  form  that  was  used  in  comput- 
ing the  corrections  5i,  ^2,  53,  etc.  If  these  residuals  are  of  the  same 
order  of  magnitude  as  the  accidental  errors  of  the  observations 
and  distributed  in  accordance  with  the  laws  of  such  errors,  the 
functional  relation 

y  =  F(A",B",C",  .  .  .  ,x,z,  .  .  .  )  (158) 

is  the  most  probable  result  that  can  be  derived  from  the  given 
observations. 

Frequently  the  residuals  corresponding  to  the  second  approxi- 
mations do  not  atisfy  the  above  conditions.  This  may  be  due 
to  the  inadequacy  of  the  assumed  form  of  the  function  F,  to 
insufficient  precision  of  the  approximations  A",  B",  C",  etc.,  or 
to  both  of  these  causes. 

If  the  form  of  the  function  is  faulty,  the  residuals  usually  show 
systematic  and  easily  recognizable  deviations  from  the  distribu- 
tion characteristic  of  accidental  errors.  Generally  the  number  of 
sign  follows  greatly  exceeds  the  number  of  sign  changes,  when  the 
residuals  are  arranged  in  the  order  of  increasing  y's,  and  opposite 
signs  do  not  occur  with  nearly  the  same  frequency.  Sometimes 
the  nature  of  the  fault  can  be  determined  by  inspecting  the  order 


ART.  91]  RESEARCH  209 

of  sequence  of  the  residuals  or  by  comparing  the  graph  correspond- 
ing to  equation  (158)  with  the  plotted  observations.  After  the 
form  of  the  function  F  has  been  rectified,  by  the  above  means  or 
otherwise,  the  computations  must  be  repeated  from  the  beginning 
and  the  new  form  must  be  tested  in  the  same  manner  as  its  prede- 
cessor. This  process  should  be  continued  until  the  residuals  cor- 
responding to  the  second  approximations  give  no  evidence  that 
the  form  of  the  function  on  which  they  depend  is  faulty. 

When  the  residuals,  computed  by  equation  (xii),  do  not  suggest 
that  the  assumed  form  of  the  function  F  is  inadequate,  but  are 
large  in  comparison  with  the  probable  errors  of  the  observations, 
the  second  approximations  are  not  sufficiently  exact.  In  such 
cases  new  equations  in  the  form  of  (157)  are  derived  by  using  A", 
B" ,  C",  etc.,  in  place  of  A',  Bf,  C',  etc.,  in  equations  (156).  The 
solution  of  the  equations  thus  formed,  by  the  method  of  least 
squares,  gives  the  corrections  5/,  52',  53',  etc.,  that  must  be  applied 
to  A",  B",  C",  etc.,  in  order  to  obtain  the  third  approximations 

At  tt          A  n    I    x  / .       T>itt         ~Dir    I     <j  / .      r</n         rut    \     *  t .     4. 
=  A    -f-  di  ;     £>      =  n    +  62  ;     C      =  C    +  03  ;  etc. 

These  operations  must  be  repeated  until  the  residuals  correspond- 
ing to  the  last  approximations  are  of  the  same  order  of  magnitude 
as  the  accidental  errors  of  the  observations. 

Although  an  algebraic  expression,  that  represents  any  given 
series  of  observations  with  sufficient  precision,  can  usually  be  de- 
rived by  the  foregoing  methods,  such  a  procedure  is  by  no  means 
advisable  in  all  cases.  In  many  investigations,  a  graphical  repre- 
sentation of  the  results  leads  to  quite  as  definite  and  trustworthy 
conclusions  as  the  more  tedious  mathematical  process.  Conse- 
quently the  latter  method  is  usually  adopted  only  when  the  former 
is  inapplicable  or  fails  to  utilize  the  full  precision  of  the  observa- 
tions. In  all  cases  the  choice  of  suitable  methods  and  the  estab- 
lishment of  rational  conclusions  is  a  matter  of  judgment  and 
experience. 

91.  Publication. —  Research  does  not  become  effective  as  a 
factor  in  the  advancement  of  science  until  its  results  have  been 
published,  or  otherwise  reported,  in  intelligible  and  widely  acces- 
sible form.  It  is  the  duty  as  well  as  the  privilege  of  the  investiga- 
tor to  make  such  report  as  soon  as  he  has  arrived  at  definite 
conclusions.  But  nothing  could  be  more  inadvisable  or  untimely 
than  the  premature  publication  of  observations  that  have  not  been 
thoroughly  discussed  and  correlated  with  fundamental  principles. 


210  THE  THEORY  OF  MEASUREMENTS       [ART.  91 

Until  an  investigation  has  progressed  to  such  a  point  that  it  makes 
some  definite  addition  to  existing  ideas,  or  gives  some  important 
physical  constant  with  increased  precision,  its  publication  is  likely 
to  retard  rather  than  stimulate  the  progress  of  science.  On  the 
other  hand,  free  discussion  of  methods  and  preliminary  results  is 
an  effective  molder  of  ideas. 

The  form  of  a  published  report  is  scarcely  less  important  than 
the  substance.  The  significance  of  the  most  brilliant  ideas  may 
be  entirely  masked  by  faulty  or  inadequate  expression.  Hence 
the  investigator  should  strive  to  develop  a  lucid  and  concise  style 
that  will  present  his  ideas  and  the  observations  that  support 
them  in  logical  sequence.  Above  all  things  he  should  remember 
that  the  value  of  a  scientific  communication  is  measured  by  the 
importance  of  the  underlying  ideas,  not  by  its  length. 

The  author's  point  of  view,  the  problem  he  proposes  to  solve, 
and  the  ideas  that  have  guided  his  work  should  be  clearly  defined. 
Theoretical  considerations  should  be  rigorously  developed  in  so 
far  as  they  have  direct  bearing  on  the  work  in  hand.  But  general 
discussions  that  can  be  found  in  well-known  treatises  or  in  easily 
accessible  journals  should  be  given  by  reference,  and  the  formulae 
derived  therein  assumed  without  further  proof  whenever  their 
rigor  is  not  questioned.  However,  the  author  should  always 
explain  his  own  interpretation  of  adopted  formulae  and  point  out 
their  significance  with  respect  to  his  observations.  Due  weight 
and  credit  should  be  given  to  the  ideas  and  results  of  other  workers 
in  the  same  or  closely  related  fields,  but  lengthy  descriptions  of 
their  methods  and  apparatus  should  be  avoided.  Explicit  refer- 
ence to  original  sources  is  usually  sufficient. 

The  methods  and  apparatus  actually  used  in  making  the  re- 
ported observations,  should  be  concisely  described,  with  the  aid 
of  schematic  diagrams  whenever  possible.  Well-known  methods 
and  instruments  should  be  described  only  in  so  far  as  they  have 
been  modified  to  fulfill  special  purposes.  Detailed  discussion  of 
all  of  the  methods  and  instruments  that  have  been  found  to  be 
inadequate  are  generally  superfluous,  but  the  difficulties  that  have 
been  overcome  should  be  briefly  pointed  out  and  explained.  The 
precautions  adopted  to  avoid  constant  errors  should  be  explicitly 
stated  and  the  processes  by  which  unavoidable  errors  of  this 
type  have  been  removed  from  the  measurements  should  be  clearly 
described.  The  effects  likely  to  arise  from  such  errors  should  be 


ART.  91]  RESEARCH  211 

considered  briefly  and  the  magnitude  of  applied  corrections  should 
be  stated. 

Observations  and  the  results  derived  from  them  should  be 
reported  in  such  form  that  their  significance  is  readily  intelligible 
and  their  precision  easily  ascertainable.  In  many  cases  graphical 
methods  of  representation  are  the  most  suitable  provided  the 
points  determined  by  the  observations  are  accurately  located 
and  marked.  The  reproduction  of  a  large  mass  of  numerical  data 
is  thus  avoided  without  detracting  from  the  comprehensiveness 
of  the  report.  When  such  methods  do  not  exhibit  the  full  pre- 
cision of  the  observations  or  when  they  are  inapplicable  on  account 
of  the  nature  of  the  problem  in  hand,  the  original  data  should  be 
reproduced  with  sufficient  fullness  to  substantiate  the  conclusions 
drawn  from  them.  In  such  cases  the  significance  of  the  obser- 
vations and  derived  results  can  generally  be  most  convincingly 
brought  out  by  a  suitable  tabulation  of  numerical  data.  An 
estimate  of  the  precision  attained  should  be  made  whenever  the 
results  of  the  investigation  can  -be  expressed  numerically. 

Final  conclusions  should  be  logically  drawn,  explicitly  stated, 
and  rigorously  developed  in  their  theoretical  bearings.  They 
express  a  culmination  of  the  author's  ideas  relative  to  the  inves- 
tigated phenomena  and  invite  criticism  of  their  exactness  and 
rationality.  Unless  they  are  amply  substantiated  by  the  obser- 
vations and  theoretical  considerations  brought  forward  in  their 
support,  and  constitute  a  real  addition  to  scientific  knowledge, 
they  are  likely  to  receive  scant  recognition. 


TABLES. 

The  following  tables  contain  formulae  and  numerical  data  that 
will  be  found  useful  to  the  student  in  applying  the  principles 
developed  in  the  preceding  chapters.  The  four  figure  numerical 
tables  are  amply  sufficient  for  the  computation  of  errors,  but  more 
extensive  tables  should  be  used  in  computing  indirectly  measured 
magnitudes  whenever  the  precision  of  the  observations  warrants 
the  use  of  more  than  four  significant  figures. 

The  references  placed  under  some  of  the  tables  indicate  the 
texts  from  which  they  were  adapted. 


TABLE  I.  —  DIMENSIONS  OF  UNITS. 


Units. 

Dimensions. 

Fundamental. 

Length,  mass,  time 

Length,  force,  time. 

Length  

[L] 
[M] 
[T] 
[LMT-*] 

m 

M 

[L-W] 
[Llr*\ 

[LT-i] 

pNj 

[LT-*\ 
[T-2] 
[LMT~l] 
[L*M] 
[LW77-1] 
[L*MT-*] 
[L-W71-2] 
[LW77-2] 

[Lwr-8] 

[L] 
[L-iFT*] 
[T] 
[F] 
[V] 
[If] 
[L-*FT*] 
[LL-i] 
[LT-i] 
[T-1] 
[LT-*] 
[T-2] 

[FT] 
[LFT*] 
[LFT] 
[LF] 
[L-*F] 
[LF] 
[LFT-1] 

Mass  

Time  

Force  

Area  

Volume  

Density  

Angle  

Velocity,  linear  

Velocity,  angular  

Acceleration,  linear  

Acceleration,  angular  

Momentum  

Moment  of  inertia  

Moment  of  momentum  

Torque  

Pressure  

Energy,  work  

Power  

212 


TABLES 


213 


TABLE  II.  —  CONVERSION  FACTORS. 


Length  Units. 

Logarithm. 

1  centimeter  (cm.)   _  =  0. 393700  inch 1 . 5951654 

"                     "  =  0. 0328083  foot 2. 5159842 

"                        =  0. 0109361  yard 2. 0388629 

1  meter  (m.)                =  1000  millimeters 3. 0000000 

"                            =  100  centimeters 2. 0000000 

"                            =  10  decimeters. 1.0000000 

1  kilometer  (km.)        =  1000  meters 3. 0000000 

=  0. 621370  mile 1. 7933503 

"                         =  3280. 83  feet 3. 5159842 

1  inch  (in.)                   =  2. 540005  centimeters 0. 4048346 

1  foot  (ft.)                   =12  inches 1.0791812 

=  30. 4801  centimeters 1 . 4840158 

1  yard  (yd.)                 =  36  inches 1. 5563025 

"                            =3  feet 0. 4771213 

"                             =  91.4402  centimeters 1.9611371 

1  mile  (ml.)                  =  5280  feet 3 . 7226339 

"                              =  1760  yards 3. 2455127 

=  1609. 35  meters 3.2066497 

=  0.868392  knot  (U.  S.) 1.9387157 

Mass  Units. 

1  gram  (g.)                  =  1000  milligrams 3. 0000000 

"                             =  100  centigrams 2. 0000000 

"                             =  10  decigrams 1. 0000000 

=  0.0352740  ounce  (av.) 2.5474542 

"                             =  0. 00220462  pound  (av.) 3. 3433342 

=  0. 000068486  slugg 5. 8355997 

1  kilogram  (kg.)          =  1000  grams 3. 0000000 

1  ounce  (oz.)  (av.)      =  28. 3495  grams 1. 4525458 

=  0. 062500  pound  (av.) 2. 7958800 

"                            =0.0019415  slugg 3.2881455 

1  pound  (Ib.)  (av.)      =  16  ounces  (av.) 1.2041200 

"                           =  453. 5924277  grams 2. 6566658 

=  0.0310646  slugg 2.4922655 

1  slugg  (sg.)                 =  32. 191  pounds  (av.) 1. 5077345 

=  515.06  ounces  (av.) 2.7118545 

=  14601.  6  grams 4. 1644003 

1  short  ton  (tn.)          =  2000  pounds  (av.) 3. 3010300 

=  907. 185  kilograms 2. 9576958 

"                        =62. 129  sluggs 1 . 7932955 


214 


THE  THEORY  OF  MEASUREMENTS 


TABLE  II.  —  CONVERSION  FACTORS  (Concluded}. 


Force  Units. 

The  following  gravitational  units  are  expressed  in  terms  of  the  earth's 
attraction  at  London  where  the  acceleration  due  to  gravity  is  32.191  ft. /sec.2 


or  981.19  cm./sec  Logarithm. 

1  dyne  =  1 . 01917  milligram's  wt 0. 0082469 

"  =  0. 00101917  gram's  wt 3 . 0082469 

"  =2.2469  X  10-6  pound's  wt 6.3515811 

1  gram's  wt.  =  981.19  dynes 2. 9917531 

1  kilogram's  wt.          =  1000  gram's  wt 3. 0000000 

=  98. 119  X  104  dynes 5.9917531 

=  2.20462  pound's  wt 0. 3433342 

1  pound's  wt.  =0. 45359  kilogram's  wt 1 . 6566658 

=  44.506  X  104  dynes 5.6484189 

1  pound's  wt.  (local)  =  0/32.191  pound's  wt.  at  London. 

g  =  local  acceleration  due  to  gravity  in  ft./secT2. 

Mean  Solar  Time  Units. 

1  second  (s.)  =  0. 016667  minute 2. 2218487 

"  =  0. 00027778  hour 4. 4436975 

=  0.000011574  day 5.0634863 

1  minute  (m.)  =  60  seconds 1 . 7781513 

"  =0.016667  hour 2.2218487 

=  0.00069444  day 4.8416375 

1  hour  (h.)  =  3600  seconds 3.  5563025 

=  60  minutes 1. 7781513 

"  =  0. 041667  day 2. 6197888 

1  day  (d.)  =  86400  seconds 4. 9365137 

=  1440  minutes 3. 1583625 

"  =24  hours 1.3802112 

1  mean  solar  unit        =  1 . 00273791  sidereal  units 0. 0011874 

Angle  Units. 

1  circumference  =  360  degrees 2. 5563025 

=  2  TT  radians 0. 7981799 

"  =  6.28319  radians 0. 7981799 

1  degree  (°)  =  0. 017453  radian 2. 2418774 

=  60  minutes 1.  7781513 

=  3600  seconds 3.  5563025 

1  minute  (')  =2. 9089  X  10-4  radians 4 . 4637261 

=  0.016667  degree 2.2218487 

=  60  seconds 1.  7781513 

1  second  (')  =  4.8481  X  KH5  radians 6 .  6855749 

=  2. 7778  X  10-4  degrees 4. 4436975 

=  0. 01667  minute 2. 2218487 

1  radian  =  57.29578  degrees 1. 7581226 

=  3437.7468  minutes 3. 5362739 

=  206264.8  seconds . .  5. 3144251 


TABLES  215 

TABLE  III.  —  TRIGONOMETRICAL  RELATIONS. 


a3    .   a5  t      <t\, 

sma  =  a— 777  +T?  —  •••(—!) 


(2n-l)! 


—  cos2  a  = 


1  —  cos  2  a 


2  cosec  a 


_    .    ce        a       cos  a       tan  a 

=  2  sin  ^  cos  tr 


2        2       cot  a       sec  a 
tan  a  1 


-  =  cos  a  tan  a 


Vl+tan2a        VI  +  cot2  a 
=  sin  /3  cos  (|8  —  a)  —  cos  /3  sin  (/3  —  a) 
=  cos  /3  sin  (0  +  a)  —  sin  /8  cos  (/3  +  a). 


l/l  — 
a  =y 


cos  a 
2~~ 

2  tan  a 


sin  2  a  =  2  sin  a  cos  a  =  ., 

1  +  tan2  a. 

sin2  a  =  1  —  cos2 a  =  —  \  (cos  2  a  —  1). 

sin  (a  ±  j8)  =  sin  a  cos  0  ±  cos  a  sin  0. 

sin  a  =fc  sin  /3  =  2  sin  £  (a  d=  /8)  cos  |(«  =F  0). 

sin2  a  +  sin2  /3  =  1  —  cos  (a  +  /3)  cos  (a  —  /8). 

sin2  a  —  sin2  0  =  cos2  /3  —  cos2  a  =  sin  (a  +  0)  sin  (a  —  /8). 


V  1  +  sin  a  =  sin  |  a  +  cos  £  a. 

VI  —  sin  a  =  ±  (sin  |  a  —  cos  \  a). 


cos 


cos2  i  a  —  sin2  |  a 
cot  a 


V 1  +  tan2  a       V 1  +  cot2  a 
sin  a          cot  a  1 


=  sin  a  cot  a 


COS  ^  a  = 


tan  o;      cosec  a      sec  a 
=  cos  0  cos  (a  +  /8)  +  sin  0  sin  (a  +  0) 
=  cos  /?  cos  ((8  -  a)  +  sin  ft  sin  (0  —  a). 

1  +  cos  a 


216  THE  THEORY  OF  MEASUREMENTS 

TABLE  III.  —  TRIGONOMETRICAL  RELATIONS  (Continued). 


cos  2  a  =  2  cos2  a  —  1  =  1  —  2  sin2  a 

1  -  tan2  a 

=  cos2  a  —  sm2  a.  =  ^ —  — 

1  +  tan2  a 

cos2  a.  =  1  -  sin2  a  =  £  (cos  2  a  +  1). 

cos  (a  d=  0)  =  cos  a  cos  0  T  sin  a  sin  0. 

cos  a  +  cos  0  =  2  cos  5  (a  +  0)  cos  H«  —  0)- 

cos  a  —  cos  0  =  —  2  sin  £  (a  -f  0)  sin  |  (a  —  0) . 

cos2  a  +  cos2  0  =  1  +  cos  (a  +  0)  cos  (a  -  0). 

cos2  a  —  cos2  0  =  sin2  0  —  sin2  a  =  —  sin  (a  +  0)  sin  (a  —  0). 

cos2  a  —  sin2  0  =  cos  (a  +  0)  cos  (a  —  0)  =  cos2  0  —  sin2  a. 

sin  a  +  cos  a  =  V 1  +  sin  2  a. 

sin  a  —  cos  a  =  Vi  —  sin 2 a. 

sin2  a  +  cos2  a.  =  1. 

sin2  a  —  cos2  a:  =  —  cos  2  or. 

tan  a  =  a  +  |  a3  +  -r25  CK5  +  3^5  a7  +  .  .   .  w >  a>  —  TT 


sin  a.          sin  2  a  1  —  cos  2 


cos  a       1  +  cos  2  a  sin  2  a 


V'l  —  cos  2  a  _  4  / 
1+  cos  2  a  "  V 


cos2  a  VI  —  sin2  a 


=  Vsec2  a  —  I 


tan  2  a  = 


cosec  a:  Vcosec2a-l 

— —  =  cot  a  —  2  cot  2  a 
cot  a 

sin  (a  +  0)  +  sin  («  —  0)    _  cos  (a  —  0)  —  cos  (a  +  0) 
cos  (a  +  0)  +  cos  (a  —  0)       sin  (a  +  0)  -  sin  (a  -  0) 

2  tan  a  2  cot  a  2 


1  —  tan2  a        cot2  a  —  1        cot  a  —  tan  a 


tan  f  a.  =  -  —  ;  -  =  cosec  a  —  cot  or. 
1  +  sec  a 

(     ±.  R\  —    tan  a  ±  tan  0    _  cos  2  0  —  cos  2  a 
*  W  *  1  T  tan  a  tan  0  ~  sin  2  0  =F  sin  2  a 

sin  (a  ±  0) 


tan  a  ±  tan  0  = 


cos  a  cos  0 


TABLES  217 

TABLE  III. — TRIGONOMETRICAL  RELATIONS  (Concluded). 


Ill  2 

cot  a.  =  --  -  a  —  j=  a3  —  ^r-=  a5  —   •  •  •  TT  >  a  >  —  IT 
a:       3  45  olo 


cos  a  _      sin  2  a      _  1  +  cos  2  a 

sin  a  ~~  1  —  cos  2  a  "~      sin  2  a 


V/ 


1  +  cos  2  o:  _        cos  a  vl  —  sin2  a 


1  —  cos  2  a       Vl  —  cos2  a 


=  tan  a.  +  2  cot  2  a. 
tan  a. 

_  1  —  tan2  a.  _  cot2  a  —  1       cot  a  —  tan  a 
"    2  tan  a  2cota  ~^~ 

cot  -  a  =  (1  +  sec  a)  cot  a 


2<-«.    —     v  j.       |      kjv/v;  <-*. y   vv/u    c*.    —  : 

cosec  a  —  cot  a 

1  =F  tan  a  tan  /?       cot «  cot  £  =F  1 


cot  (a  d=  0)  = 


tan  a  ±  tan  0         cot  0  d=  cot  a 


sin 


TABLE  IV.  —  SERIES. 


Taylor's  Theorem. 

/(*+&)=/(*)  +  AT  (*)  +  ^/"  (*)+••;+  ^/W  (x)  + 

f(x  +  h,     y  +  k, 


where  u  =  f  (x,  y,  z). 

Maclaurin's  Series. 


/(0)  +  f  /'  (0)  +  !/"  (0)  +  •  •  •  +  fj/N  (0). 


218  THE   THEORY  OF  MEASUREMENTS 

TABLE  IV.  —  SERIES  (Concluded}. 


Binomial  Theorem. 

=  xm  +  rnx^ly  +  m(n^xm_^  + 

.  .  .    ,  *»  (m  -  1)  .  .  .  (m  -  n  +  1)  ^- y> 

when  m  is  a  positive  integer,  also  when  m  is  negative  or  fractional  and 
x  >  y.  When  x  <  y  and  m  is  fractional  or  negative  the  series  must  be 
taken  in  the  form 


(x  +  y)m  =  ym  +  j  ym-*x+      v^     *'  y*-'z»  +  •  •  • 

m  (m  -  1)  .  .  .  (m  -  n  +  1) 

n! 
Fourier's  Series. 

j-  /     \  It        it  ""•E      i      t  2  7TX  ,  3  7TX      . 

/  (x)  =  -  60  +  &i  cos H  &2  cos  —  -  +  63  cos H  •  •  • 

£  C  C  C 

.     TTX    .  .     2irX    .  .     STTX    , 

+  01  sin \-  a2  sin h  a3  sin f-  •  •  • 

c  c  c 

where 


1    r  +  c,/   v          WTTX   . 
>m  =  ~    I         / (*)  COS  -—  dx, 

C    •/  —  c  t/ 

1    f+cr/  v    .    m-n-x  , 
m  =  -   \      f(x)  sm  — —  dx, 

C    •/  —  c  ^ 


2    /»c  ,  ,   ,     .     WTTX  , 

=  -   I    /  W  sin  -  "£• 

C   «/o  C 


provided  /  (x)  is  single  valued,  uniform,  and  continuous,  and  c  >  x  > 
—  c.  For  values  of  x  lying  between  zero  and  c  the  function  may  be  ex- 
panded in  the  form 

,  /    x  .      TTX     .  .      2-JTX      .  .       3  TTX     , 

f  (x)  =  0,1  sin  --  \-a-i  sin  --  H  a3  sin  ---  (-  •  •  •  , 
where  a 

Also      f(x)  =^60  +  61  cosy  4-62cos         +  63cos 

2    rc  -  /   x          WTTX  , 

where  bm  =  -  I    /  (x)  cos  -  ax. 

C  JQ  C 

General  Series. 


xloga       (x  log  a)2       (x  log  a)3  (x  log  a)n 

~~     ~~      ~~  —~ 


.-  :»>} 


TABLES 


219 


TABLE  V.  —  DERIVATIVES. 
U,  F,  W  any  functions;  a,  6,  c  constants. 


dx 


F2 


S:***St^T? 


axx 


a  ,  logae. 

_logax=— , 


dU 


a  .  i  at; 

_logaC7.=  __ 


V  dx 

=  ax  log  a. 


dx 

d 
dx( 

a 

dx 


—  sm  x  =  cos  x. 
ax 


.  r,      . 

—  sm  aC7  =  a  cos  ac7  ^—  , 
ax  ax 

a  l 

—  tan  x  =  —  r—  =  sec2  x: 
ax  cos2  x 


—  cos  x  =  —  sm  x. 
ax 

a  -i 

—  cot  x  =    .  ,     =  —  cosec2  x. 
ax  sin2  x 


—  sec  x  =  tan  x  sec  x; 

oX 


—  cosec  x  =  —  cot  x  cosec  x. 
ax 


—  log  sinx  =  cotx; 


—  log  cos  x  =  —  tan  x. 

ox 


The  following  expressions  for  the  derivatives  of  inverse  functions  hold 
for  angles  in  the  first  and  third  quadrants.  For  angles  in  the  second  and 
fourth  quadrants  the  signs  should  be  reversed. 


ax 

—  tan-1  x  =  •= 

ax  i 


. 

T-  cos-1  x  = 
dx 


i 


220  THE  THEORY  OF  MEASUREMENTS 

TABLE  VI.  —  SOLUTION  OF  EQUATIONS. 

The  following  algebraic  expressions  for  the  roots  of  equations  of  the 
second,  third,  and  fourth  degrees  are  in  the  form  given  by  Merriman. 
(Merriman  and  Woodward,  "Higher  Mathematics";  Wiley  and  Sons, 
1896.) 

The  Quadratic  Equation. 
Reduce  to  the  form 

x2 
Then  the  two  roots  are 


x\  =  —  a  +     a?  —  6;    z2  =  —  a  —  Va2  —  b 

The  Cubic  Equation. 
Reduce  to  the  form 

=  0. 


Compute  the  following  auxiliary  quantities  : 

B  =  -  a2  +  6;  C  =  a3  -  f  ab  +  c; 


Then  the  three  roots  are 

xi=-a  +  (si  +  s2),  _ 

xz=-a  -M«i+s2)  +|  V-_3(Sl  -s2), 
x3  =  -  a  -  HSI  +  s2)  -  |  V-  3  (si  -  sa). 

When  B3  +  C2  is  negative  the  roots  are  all  real  but  they  cannot  be  de- 
termined numerically  by  the  above  formulae  owing  to  the  complex  nature 
of  si  and  s2.  In  such  cases  the  numerical  values  of  the  roots  can  be  deter- 
mined only  by  some  method  of  approximation. 

The  Quartic  Equation. 
Reduce  to  the  form 

z4  +  4az3  +  66z2  +  4cz  +  d  =  0. 
Compute  the  following  auxiliary  quantities  : 

g  =  a*-b;    h  =  63  +  c2-2abc  +  dg;     fc  =  |ac  -  62  -  |d; 

I  =  I  (h  +  V^TF')*  +  1  (h  -  VF+^)*; 

u  =  g  +  l',    v  =  2g-l;     w  =  4u*  +  3k  -  12gl. 
Then  the  four  roots  are 

xi  =  —  a  +  ^u  +  Vy  + 


—  a  —     u  — 


in  which  the  signs  are  to  be  used  as  written  provided  that  2  a3  —  3  ab  +  c 
is  a  negative  number;  but  if  this  is  positive  all  radicals  except  Vw  are  to 
be  changed  in  sign. 

The  above  expressions  are  irreducible  when  hz  +  k*  is  a  negative  number. 
In  this  case  the  given  equation  has  either  four  imaginary  roots  or  four  real 
roots  that  can  be  determined  numerically  only  by  some  method  of  approxi- 
mation. 


TABLES  221 

TABLE  VII.  —  APPROXIMATE  FORMULA. 

In  the  following  formulae,  a,  /3,  5,  etc.,  represent  quantities  so  small  that 
their  squares,  higher  powers,  and  products  are  negligible  in  comparison  with 
unity.  The  limit  of  negligibility  depends  on  the  particular  problem  in 
hand.  Most  of  the  formulae  give  results  within  one  part  in  one  million 
when  the  variables  are  equal  to  or  less  than  0.001. 

1.    (l+a)n=l+n«;     (1  -a)n  =  1  -  na. 


4. 


6          l        =  1  --'          ,l        =  1  +-• 
'    Vl+«  n'        Vl  -a  n 

7. 


9.    (x  +  a 

When  the  angle  a,  expressed  in  radians,  is  small  in  comparison  with  unity 
a  first  approximation  gives 

10.  sin  a  =  a',     sin  (x  ±  a)  =  sin  x  ±  a  cos  x. 

11.  cos  a  =  1;     cos  (x  ±  a)  =  cos  x  =F  a  sin  x. 

12.  tan  a  =  a]     tan  (x  d=  a)  =  tana;  ± — ^— • 
The  second  approximation  gives 


13.  sin  a  =  a  —  -TT  ;  sin2  a  =  a2    1  —  ^r- 

o  \         o 

a2 

14.  cos  a  =  1  —  -5- ;  COS2  a  =  1  —  a2. 

3  /          o      \ 

15.  tana  =  a  +  ^-|  tan2a  =  a2  (  1  +  ^  a2  V 

(Kohlrausch,  "Praktische  Physik.") 


222  THE  THEORY  OF  MEASUREMENTS 


TABLE  VIII.  —  NUMERICAL  CONSTANTS. 


Logarithm  . 

Base  of  Naperian  logarithms:  e  =  2.  7182818  ........     0.  4342945 

Modulus  of  Naperian  log.:  M  =  ^  =  2.30259  ...........     0.3622157 

Modulus  of  common  log.:         =  log  e  =  0.  4342945  .........     1.  6377843 


Circumference  ,..  14159265  .  0.  4971499 


Diameter 

2?r  =  6.28318530  ..............  0.7981799 

-  =0.3183099  .  1.5028501 

7T 

Tr2  =  9.8696044  .  .  .............  0.9942998 

V^  =  1.7724539  ...............  0.2485749 

|  =  0.7853982  ...............  1.8950899 

5  =0.5235988  .  1.7189986 
o 

w  =  Precision  constant;    k  =  Unit  error;    A  =  Average  error; 
M  =  Mean  error;     E  =  Probable  error. 

4p  =  0.31831  .................  1.5028501 

^  =  0.39894  .................  1.6009101 

^  =  0.26908  .................  1.4298888 

^  =  1.25331   .................  0.0980600 

A. 

f  =  0.84535   .................  1.9270387 

A. 

=  0.67449  .................  1.8289787 


TABLES 


223 


TABLE  IX.  —  EXPONENTIAL  FUNCTIONS. 


X 

logic  (e*) 

e* 

e* 

X 

log  10  (O 

e' 

e~' 

0.0 

0.00000 

1.0000 

1.000000 

5.0 

2.17147 

148.41 

0.006738 

0.1 

0.04343 

1.1052 

0.904837 

5.1 

2.21490 

164.02 

0.006097 

0.2 

0.08686 

1.2214 

0.818731 

5.2 

2.25833 

181.27 

0.005517 

0.3 

0.13029 

1.3499 

0.740818 

5.3 

2.30176 

200.34 

0.004992 

0.4 

0.17372 

1.4918 

0.670320 

5.4 

2.34519 

221.41 

0.004517 

0.5 

0.21715 

1.6487 

0.606531 

5.5 

2.38862 

244.69 

0.004087 

0.6 

0.26058 

1.8221 

0.548812 

5.6 

2.43205 

270.43 

0.003698 

0.7 

0.30401 

2.0138 

0.496585 

5.7 

2.47548 

298.87 

0.003346 

0.8 

0.34744 

2.2255 

0.449329 

5.8 

2.51891 

330.30 

0.003028 

0.9 

0.39087 

2.4596 

0.406570 

5.9 

2.56234 

365.04 

0.002739 

1.0 

0.43429 

2.7183 

0.367879 

6.0 

2.60577 

403.43 

0.002479 

1.1 

0.47772 

3.0042 

0.332871 

6.1 

2.64920 

445.86 

0.002243 

1.2 

0.52115 

3.3201 

0.301194 

6.2 

2.69263 

492.75 

0.002029 

1.3 

0.56458 

3.6693 

0.272532 

6.3 

2.73606 

544.57 

0.001836 

1.4 

0.60801 

4.0552 

0.246597 

6.4 

2.77948 

601.85 

0.001662 

.5 

0.65144 

4.4817 

0.223130 

6.5 

2.82291 

665.14 

0.001503 

.6 

0.69487 

4.9530 

0.201897 

6.6 

2.86634 

735.10 

0.001360 

.7 

0.73830 

5.4739 

0.182684 

6.7 

2.90977 

812.41 

0.001231 

.8 

0.78173 

6.0496 

0.165299 

6.8 

2.95320 

897.85 

0.001114 

.9 

0.82516 

6.6859 

0.149569 

6.9 

2.99663 

992.27 

0.001008 

2.0 

0.86859 

7.3891 

0.135335 

7.0 

3.04006 

1096.6 

0.000912 

2.1 

0.91202 

8.1662 

0.122456 

7.1 

3.08349 

1212.0 

0.000825 

2.2 

0.95545 

9.0250 

0.110803 

7.2 

3.12692 

1339.4 

0.000747 

2.3 

0.99888 

9.9742 

0.100259 

7.3 

3.17035 

1480.3 

0.000676 

2.4 

1.04231 

11.023 

0.090718 

7.4 

3.21378 

1636.0 

0.000611 

2.5 

1.08574 

12.182 

0.082085 

7.5 

3.25721 

1808.0 

0.000553 

2.6 

1.12917 

13.464 

0.074274 

7.6 

3.30064 

1998.2 

0.000500 

2.7 

1  .  17260 

14.880 

0.067206 

7.7 

3.34407 

2208.3 

0.000453 

2.8 

1.21602 

16.445 

0.060810 

7.8 

3.38750 

2440.6 

0.000410 

2.9 

1.25945 

18.174 

0.055023 

7.9 

3.43093 

2697.3 

0.000371 

3.0 

1.30288 

20.086 

0.049787 

8.0 

3.47436 

2981.0 

0.000335 

3.1 

1.34631 

22.198 

0.045049 

8.1 

3.51779 

3294.5 

0.000304 

3.2 

1.38974 

24.533 

0.040762 

8.2 

3.56121 

3641.0 

0.000275 

3.3 

1.43317 

27.113 

0.036883 

8.3 

3.60464 

4023.9 

0.000249 

3.4 

1.47660 

29.964 

0.033373 

8.4 

3.64807 

4447.1 

0.000225 

3.5 

1.52003 

33.115 

0.030197 

8.5 

3.69150 

4914.8 

0.000203 

3.6 

1.56346 

36.598 

0.027324 

8.6 

3.73493 

5431.7 

0.000184 

3.7 

1.60689 

40.447 

0.024724 

8.7 

3.77836 

6002.9 

0.000167 

3.8 

1.65032 

44.701 

0.022371 

8.8 

3.82179 

6634.2 

0.000151 

3.9 

1.69375 

49.402 

0.020242 

8.9 

3.86522 

7332.0 

0.000136 

4.0 

1.73718 

54.598 

0.018316 

9.0 

3.90865 

8103.1 

0.000123 

4.1 

.78061 

60.340 

0.016573 

9.1 

3.95208 

8955.3 

0.000112 

4.2 

.82404 

66.686 

0.014996 

9.2 

3.99551 

9897.1 

0.000101 

4.3 

.86747 

73.700 

0.013569 

9.3 

4.03894 

10938. 

0.000091 

4.4 

.91090 

81.451 

0.012277 

9.4 

4.08237 

12088. 

0.000083 

4.5 

.95433 

90.017 

0.011109 

9.5 

4.12580 

13360. 

0.000075 

4.6 

.99775 

99.484 

0.010052 

9.6 

4.16923 

14765. 

0.000068 

4.7 

2.04118 

109.95 

0.009095 

9.7 

4.21266 

16318. 

0.000061 

4.8 

2.08461 

121.51 

0.008230 

9.8 

4.25609 

18034. 

0.000055 

4.9 

2.12804 

134.29 

0.007447 

9.9 

4.29952 

19930. 

0.000050 

5.0 

2.17147 

148.41 

0.006738 

10.0 

4.34294 

22026. 

0.000045 

Taken  from  Glaisher's  "Tables  of  the  Exponential  Function,"  Trans.  Cambridge  Phil.  Soc., 
vol.  xiii,  1883.  This  volume  also  contains  a  "  Table  of  the  Descending  Exponential  to  Twelve 
or  Fourteen  Places  of  Decimals,"  by  F.  W.  Newman. 


224 


THE   THEORY  OF  MEASUREMENTS 


TABLE  X.  —  EXPONENTIAL  FUNCTIONS. 
Value  of  ex<t  and  erx<i  and  their  logarithms. 


X 

<? 

log  e»2 

e~*2 

log  e'*z 

0.1 

1.0101 

0.00434 

0.99005 

1.99566 

0.2 

1.0408 

0.01737 

0.96079 

1.98263 

0.3 

1.0942 

0.03909 

0.91393 

1.96091 

0.4 

.1735 

0.06949 

0.85214 

.93051 

0.5 

.2840 

0.10857 

0.77880 

.89143 

0.6 

.4333 

0.15635 

0.69768 

.84365 

0.7 

.6323 

0.21280 

0.61263 

.78720 

0.8 

.8965 

0.27795 

0.52729 

.72205 

0.9 

2.2479 

0.35178 

0.44486 

.64822 

1.0 

2.7183 

0.43429 

0.36788 

.56571 

1.1 

3.3535 

0.52550 

0.29820 

.47450 

1.2 

4.2207 

0.62538 

0.23693 

.37462 

1.3 

5.4195 

0.73396 

0.18452 

.26604 

1.4 

7.0993 

0.85122 

0.14086 

.14878 

1.5 

9.4877 

0.97716 

0.10540 

.02284 

1.6 

1.2936X10 

1.11179 

0.  77305  XlO-1 

2.88821 

1.7 

1.7993X10 

1.25511 

0.  55576  XlO-1 

2.74489 

1.8 

2.5534x10 

1.40711 

0.  39164  XlO-1 

2.59289 

1.9 

3.6966X10 

1.56780 

0.27052  XlO-1 

2.43220 

2.0 

5.4598X10 

1.73718 

0.18316  XlO-1 

2.26282 

2.1 

8.2269x10 

1.91524 

0.12155  XlO-1 

2.08476 

22 

1.2647X102 

2.10199 

0.79071  XlO-2 

3.89801 

2.3 

1.9834X102 

2.29742 

0.50417  XlO-2 

3.70258 

2.4 

3.1735X102 

2.50154 

0.31511  XlO-2 

3.49846 

2.5 

5.1801X102 

2.71434 

0.19305  XlO-2 

3.28566 

2.6 

8.6264X102 

2.93583 

0.1  1592  XlO-2 

3.06417 

2.7 

1.4656X103 

3.16601 

0.68232X10-3 

4.83399 

2.8 

2.5402X103 

3.40487 

0.39367X10-3 

4.59513 

2.9 

4.4918X103 

3.65242 

0.22263X10-3 

4.34758 

3.0 

8.1031X103 

3.90865 

0.12341X10-3 

4.09135 

3.1 

1.4913X104 

4.17357 

0.67055x10-* 

5.82643 

3.2 

2.8001X104 

4.44718 

0.35713  XlO-4 

5.55282 

3.3 

5.3637X104 

4.72947 

0.18644  XlO-4 

5.27053 

3.4 

1.0482X105 

5.02044 

0.95403  XlO-5 

6.97956 

3.5 

2.0898X105 

5.32011 

0.47851  XlO-5 

6.67989 

3.6 

4.2507X105 

5.62846 

0.23526  XlO-5 

6.37154 

3.7 

8.8204X105 

5.94549 

0.1  1337  XlO-5 

6.05451 

3.8 

1.8673X106 

6.27121 

0.53554  XlO-6 

7.72879 

3.9 

4.0329X106 

6.60562 

0.24796  XlO-6 

?.  39438 

4.0 

8.8861X106 

6.94871 

0.11254X10-6 

7.05129 

41 

1.9975X107 

7.30049 

0.50062  XlO-7 

§.69951 

4.2 

4.5809X107 

7.66095 

0.21830  XlO-7 

S.  33905 

4.3 

1.0718X108 

8.03011 

0.93302  XlO-8 

9.96989 

4.4 

2.5582X108 

8.40794 

0.39089  XlO-8 

9.59206 

4.5 

6.2296X108 

8.79446 

0.16052X10-8 

9.20554 

4.6 

1.5476X109 

9.18967 

0.64614X10-9 

10.81033 

4.7 

3.9226X109 

9.59357 

0.25494X10-9 

10.40643 

4.8 

1.0143X1010 

10.00615 

0.98594  XlO-10 

11.99385 

4.9 

2.6755X1010 

10.42741 

0.37376  XlO-10 

11.57259 

5.0 

7.2005X1010 

10.85736 

0.13888  XlO-10 

11.14264 

TABLES 


225 


TABLE  XI.  —  VALUES  OF  THE  PROBABILITY  INTEGRAL. 


t 

P* 

Diff. 

t 

^A 

Diff. 

1 

^A 

Diff 

t 

^A 

Diff. 

0.00 

0.00000 

1  IOC 

0.50 

0.52050 

074 

1.00 

0.84270 

A  1  • 

1.50 

0.96611 

0.01 

0.01128 

1   I  —  ! 

1  100 

0.51 

0.52924 

o/  1 

O/3f> 

1.01 

0.84681 

411 
A  AO 

1.51 

0.96728 

0.02 
0.03 
0.04 
0.05 
0.06 
0.07 
0.08 
0.09 

0.02256 
0.03384 
0.04511 
0.05637 
0.06762 
0.07886 
0.09008 
0.10128 

liZo 

1128 
1127 
1126 
1125 
1124 
1122 
1120 

1  1  1C 

0.52 
0.53 
0.54 
0.55 
0.56 
0.57 
0.58 
0.59 

0.53790 
0.54646 
0.55494 
0.56332 
0.57162 
0.57982 
0.58792 
0.59594 

ODD 

856 
848 
838 
830 
820 
810 
802 

7QO 

1.02 
1.03 
1.04 
1.05 
1.06 
1.07 
1.08 
1.09 

0.85084 
0.85478 
0.85865 
0.86244 
0.86614 
0.86977 
0.87333 
0.87680 

403 
394 
387 
379 
370 
363 
356 
347 

O/M 

1.52 
1.53 
1.54 
1.55 
1.56 
1.57 
1.58 
1.59 

0.96841 
0.96952 
0.97059 
0.97162 
0.97263 
0.97360 
0.97455 
0.97546 

113 

111 

107 
103 
101 
97 
95 
91 

OA 

0.10 
0.11 
0.12 

0.11246 
0.12362 
0.13476 

1  1  J.O 

1116 
1114 
1111 

0.60 
0.61 
0.62 

0.60386 
0.61168 
0.61941 

/y^ 

782 

773 

7«j 

1.10 
1.11 
1.12 

0.88021 
0.88353 
0.88679 

O41 

332 
326 

010 

1.60 
1.61 
1.62 

0.97635 
0.97721 
0.97804 

89 
86 
83 

Qf\ 

0.13 
0.14 
0.15 
0.16 
0.17 

0.14587 
0.15695 
0.16800 
0.17901 
0.18999 

1  1  i.  i 
1108 
1105 
1101 
1098 

1  AQ~ 

0.63 
0.64 
0.65 
0.66 
0.67 

0.62705 
0.63459 
0.64203 
0.64938 
0.65663 

<  D^± 

754 
744 
735 
725 

71  t\ 

1.13 
1.14 
1.15 
1.16 
1.17 

0.88997 
0.89308 
0.89612 
0.89910 
0.90200 

OIo 

311 
304 

298 
290 

oo/i 

1.63 
.64 
.65 
.66 
.67 

0.97884 
0.97962 
0.98038 
0.98110 
0.98181 

oU 

78 
76 
72 
71 

f*Q 

0.18 
0.19 
0.20 
0.21 
0.22 
0.23 
0.24 
0.25 
0.26 
0.27 
0.28 
0.29 
0.30 
0.31 
0.32 

0.20094 
0.21184 
0.22270 
0.23352 
0.24430 
0.25502 
0.26570 
0.27633 
0.28690 
0.29742 
0.30788 
0.31828 
0.32863 
0.33891 
0.34913 

iuyo 
1090 
1086 
1082 
1078 
1072 
1068 
1083 
1057 
1052 
1046 
1040 
1035 
1028 
1022 

1  A1  K. 

0.68 
0.69 
0.70 
0.71 
0.72 
0.73 
0.74 
0.75 
0.76 
0.77 
0.78 
0.79 
0.80 
0.81 
0.82 

0.66378 
0.67084 
0.67780 
0  .  68467 
0.69143 
0.69810 
0.70468 
0.71116 
0.71754 
0.72382 
0.73001 
0.73610 
0.74210 
0.74800 
0.75381 

<  10 
706 
696 
687 
676 
667 
658 
648 
638 
628 
619 
609 
600 
590 
581 

CT1 

1.18 
1.19 
1.20 
1.21 
1.22 
1.23 
1.24 
1.25 
1.26 
1.27 
1.28 
1.29 
1.30 
1.31 
1.32 

0.90484 
0.90761 
0.91031 
0.91296 
0.91553 
0.91805 
0.92051 
0.92290 
0.92524 
0.92751 
0.92973 
0.93190 
0.93401 
0.93606 
0.93807 

Zo4 

277 
270 
265 
257 
252 
246 
239 
234 
227 
222 
217 
211 
205 
201 

.68 
.69 
.70 
.71 
.72 
.73 
1.74 
1.75 
1.76 
1.77 
1.78 
1.79 
1.80 
1.81 
1.82 

0.98249 
0.98315 
0.98379 
0.98441 
0.98500 
0.98558 
0.98613 
0.98667 
0.98719 
0.98769 
0.98817 
0.98864 
0.98909 
0.98952 
0.98994 

Do 

66 
64 
62 
59 
58 
55 
54 
52 
50 
48 
47 
45 
43 
42 

0.33 
0.34 
0.35 
0.36 
0.37 
0.38 
0.39 

0.35928 
0.36936 
0.37938 
0.38933 
0.39921 
0.40901 
0.41874 

lUio 
1008 
1002 
995 
988 
980 
973 

Qf?r 

0.83 
0.84 
0.85 
0.86 
0.87 
0.88 
0.89 

0.75952 
0.76514 
0.77067 
0.77610 
0.78144 
0.78669 
0.79184 

571 
562 
553 
543 
534 
525 
515 

CA*7 

.33 
.34 
.35 
.36 
.37 
.38 
.39 

0.94002 
0.94191 
0.94376 
0.94556 
0.94731 
0.94902 
0.95067 

195 
189 
185 
180 
175 
171 
165 

1  £JO 

1.83 
1.84 
1.85 
1.86 
1.87 
1.88 
1.89 

0.99035 
0.99074 
0.99111 
0.99147 
0.99182 
0.99216 
0.99248 

41 
39 
37 
36 
35 
34 

32 
01 

0.40 
0.41 

0.42839 
0.43797 

yoo 

958 

nr:n 

0.90 
0.91 

0.79691 
0.80188 

oU/ 
497 

4P.Q 

.40 
.41 

0.95229 
0.95385 

ItW 

156 
i  ^°. 

1.90 
1.91 

0.99279 
0.99309 

ol 

30 

on 

0.42 

0.44747 

you 
n/io 

0.92 

0.80677 

"±oy 

A>-r{\ 

1.42 

0.95538 

100 

1  AO 

1.92 

3.99338 

zy 

oo 

0.43 
0.44 

0.45689 
0.46623 

y4z 
934 

QOC 

0.93 
0.94 

0.81156 
0.81627 

479 
471 
4fi9 

1.43 
1.44 

0.95686 
0.95830 

148 
144 
140 

1.93 
1.94 

3.99366 
3.99392 

28 
26 

9fi 

0.45 

0.47548 

t/^O 
QIC 

0.95 

0.82089 

'±\j£ 
4CQ 

1.45 

0.95970 

J.TAJ 

IOC 

1.95 

3.99418 

^O 

OK 

0.46 
0.47 
0.48 
0.49 
0.50 

0.48466 
0.49375 
0.50275 
0.51167 
0.52050 

«7  j.o 

909 
900 
892 
883 

0.96 
0.97 
0.98 
0.99 
1.00 

0.82542 
0.82987 
0.83423 
0.83851 
0.84270 

^too 
445 
436 

428 
419 

1.46 
1.47 
1.48 
1.49 
1.50 

0.96105 
0.96237 
0.96365 
0.96490 
0.96611 

J.OO 

132 

128 
125 
121 

1.96 
1.97 
1.98 
1.99 
2.00 

3.99443 
3.99466 
3.99489 
3.99511 
3.99532 

^O 

23 
23 
22 
21 

oo 

.00000 

(Chauvenet,  "  Spherical  and  Practical  Astronomy.") 


226  THE  THEORY  OF  MEASUREMENTS 

TABLE  XII.  —  VALUES  OF  THE  PROBABILITY  INTEGRAL. 


3 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 

.00000 

.00538 

.01076 

.01614 

.02152 

.02690 

.03228 

.03766 

.  04303 

.04840 

0.1 

.05378 

.05914 

.06451 

.06987 

.07523 

.08059 

.08594 

.09129 

.09663 

.  10197 

0.2 

.10731 

.11264 

.11796 

.  12328 

.  12860 

.  13391 

.  13921 

.  14451 

.  14980 

.  15508 

0.3 

.16035 

.  16562 

.  17088 

.  17614 

.  18138 

.  18662 

.19185 

.  19707 

.20229 

.20749 

0.4 

.21268 

.21787 

.22304 

.22821 

.23336 

.23851 

.24364 

.24876 

.25388 

.25898 

0.5 

.26407 

.26915 

.27421 

.27927 

.28431 

.28934 

.29436 

.29936 

.30435 

.30933 

0.6 

.31430 

.31925 

.32419 

.32911 

.33402 

.33892 

.34380 

.34866 

.35352 

.35835 

0.7 

.36317 

.36798 

.37277 

.37755 

.38231 

.38705 

.39178 

.39649 

.40118 

.40586 

0.8 

.41052 

.41517 

.41979 

.42440 

.  42899 

.  43357 

.  43813 

.  44267 

.44719 

.45169 

0.9 

.45618 

.46064 

.46509 

.46952 

.47393 

.  47832 

.  48270 

.  48605 

.49139 

.49570 

.0 

.50000 

.50428 

.50853 

.51277 

.51699 

.52119 

.52537 

.52952 

.53366 

.53778 

.1 

.54188 

.54595 

.55001 

.55404 

.55806 

.56205 

.56602 

.  56998 

.57391 

.57782 

.2 

.58171 

.58558 

.58942 

.59325 

.59705 

.60083 

.60460 

.  60833 

.61205 

.61575 

.3 

.61942 

.62308 

.62671 

.63032 

.63391 

.63747 

.64102 

.64454 

.64804 

.65152 

.4 

.65498 

.65841 

.66182 

.66521 

.66858 

.67193 

.67526 

.67856 

.68184 

.68510 

.5 

.68833 

.69155 

.69474 

.69791 

.70106 

.70419 

.70729 

.71038 

.71344 

.71648 

.6 

.71949 

.72249 

.72546 

.72841 

.73134 

.73425 

.73714 

.74000 

.74285 

.74567 

.7 

.74847 

.75124 

.75400 

.75674 

.75945 

.76214 

.76481 

.76746 

.77009 

.77270 

.8 

.77528 

.77785 

.78039 

.78291 

.78542 

.78790 

.79036 

.79280 

.79522 

.79761 

.9 

.79999 

.80235 

.80469 

.80700 

.80930 

.81158 

.81383 

.81607 

.81828 

.82048 

2.0 

.82266 

.82481 

.82695 

.82907 

.83117 

.83324 

.83530 

.83734 

.83936 

.84137 

2.1 

.84335 

.84531 

.84726 

.84919 

.85109 

.85298 

.85486 

.85671 

.85854 

.86036 

2.2 

.86216 

.86394 

.86570 

.86745 

.86917 

.87088 

.87258 

.87425 

.87591 

.87755 

2.3 

.87918 

.88078 

.88237 

.88395 

.88550 

.88705 

.88857 

.89008 

:  89157 

.89304 

2.4 

.89450 

.89595 

.89738 

.89879 

.90019 

.90157 

.90293 

.90428 

.90562 

.90694 

25 

.90825 

.90954 

.91082 

.91208 

.91332 

.91456 

.91578 

.91698 

.91817 

.91935 

2.6 

.92051 

.92166 

.92280 

.92392 

.92503 

.92613 

.92721 

.92828 

.92934 

.93038 

2.7 

.93141 

.93243 

.93344 

.93443 

.93541 

.93638 

.93734 

.93828 

.93922 

.94014 

2.8 

.94105 

.94195 

.94284 

.94371 

.94458 

.94543 

.94627 

.94711 

.94793 

.94874 

2.9 

.94954 

.95033 

.95111 

.95187 

.95263 

.95338 

.95412 

.95485 

.95557 

.95628 

3 

.95698 

.96346 

96910 

.97397 

.97817 

.98176 

.98482 

.98743 

.98962 

.99147 

4 

.99302 

.99431 

.99539 

.99627 

.99700 

.99760 

.99808 

.99848 

.99879 

.99905 

5 

.99926 

.99943 

.99956 

.99966 

.99974 

.99980 

.99985 

.  99988 

.99991 

.99993 

TABLE  XIII. — CHAUVENET'S  CRITERION. 


N 

T 

N 

r 

AT 

r 

3 

2.05 

13 

3.07 

23 

3.40 

4 

2.27 

14 

3.11 

24 

3.43 

5 

2.44 

15 

3.15 

25 

3.45 

6 

2.57 

16 

3.19 

30 

3.55 

7 

2.67 

17 

3.22 

40 

3.70 

8 

2.76 

18 

3.26 

50 

3.82 

9 

2.84 

19 

3.29 

75 

4.02 

10 

2.91 

20 

3.32 

100 

4.16 

11 

2.97 

21 

3.35 

200 

4.48 

12 

3.02 

22 

3.38 

500 

4.90 

TABLES 


227 


TABLE  XTV.  —  FOR  COMPUTING  PROBABLE  ERRORS  BY  FORMULA 

(31)  AND  (32). 


AT 

0.6745 

0.6745 

AT 

0.6745 

0.6745 

iV 

VJv^T 

VN(N-l) 

iM 

vim 

v#  (AT-  i) 

40 

0.1080 

0.0171 

41 

0.1066 

0.0167 

2 

0.6745 

0.4769 

42 

0.1053 

0.0163 

3 

0.4769 

0.2754 

43 

0.1041 

0.0159 

4 

0.3894 

0.1947 

44 

0.1029 

0.0155 

5 

0.3372 

0.1508 

45 

0.1017 

0.0152 

6 

0.3016 

0.1231 

46 

0.1005 

0.0148 

7 

0.2754 

0.1041 

47 

0.0994 

0.0145 

8 

0.2549 

0.0901 

48 

0.0984 

0.0142 

9 

0.2385 

0.0795 

49 

0.0974 

0.0139 

10 

0.2248 

0.0711 

50 

0.0964 

0.0136 

11 

0.2133 

0.0643 

51 

0.0954 

0.0134 

12 

0.2029 

0.0587 

52 

0.0944 

0.0131 

13 

0.1947 

0.0540 

53 

0.0935 

0.0128 

14 

0.1871 

0.0500 

54 

0.0926 

0.0126 

15 

0.1803 

0.0465 

55 

0.0918 

0.0124 

16 

0.1742 

0.0435 

56 

0.0909 

0.0122 

17 

0.1686 

0.0409 

57 

0.0901 

0.0119 

18 

0.1636 

0.0386 

58 

0.0893 

0.0117 

19 

0.1590 

0.0365 

59 

0.0886 

0.0115 

20 

0.1547 

0.0346 

60 

0.0878 

0.0113 

21 

0.1508 

0.0329 

61 

0.0871 

0.0111 

22 

0.1472 

0.0314 

62 

0.0864 

0.0110 

23 

0.1438 

0.0300 

63 

0.0857 

0.0108 

24 

0.1406 

0.0287 

64 

0.0850 

0.0106 

25 

0.1377 

0.0275 

65 

0.0843 

0.0105 

26 

0.1349 

0.0265 

66 

0.0837 

0.0103 

27 

0.1323 

0.0255 

67 

0.0830 

0.0101 

28 

0.1298 

0.0245 

68 

0.0824 

0.0100 

29 

0.1275 

0.0237 

69 

0.0818 

0.0098 

30 

0.1252 

0.0229 

70 

0.0812 

0,0097 

31 

0.1231 

0.0221 

71 

0.0806 

0.0096 

32 

0.1211 

0.0214 

72 

0.0800 

0.0094 

33 

0.1192 

0.0208 

73 

0.0795 

0.0093 

34 

0.1174 

0.0201 

74 

0.0789 

0.0092 

35 

0.1157 

0.0196 

75 

0.0784 

0.0091 

36 

0.1140 

0.0190 

80 

0.0759 

0.0085 

37 

0.1124 

0.0185 

85 

0.0736 

0.0080 

38 

0.1109 

0.0180 

90 

0.0713 

0.0075 

39 

0.1094 

0.0175 

100 

0.0678 

0.0068 

(Merriman,  "  Least  Squares. ") 


228 


THE   THEORY  OF  MEASUREMENTS 


TABLE  XV.  —  FOR  COMPUTING  PROBABLE  ERRORS  BY  FORMULAE  (34). 


N 

0.8453 

0.8453 

N 

0.8453 

0.8453 

^N(N  -  1) 

N^N-1 

VN(N  -  1) 

N^W=1 

40 

0.0214 

0.0034 

41 

0.0209 

0.0033 

2 

0.5978 

0.4227 

42 

0.0204 

0.0031 

3 

0.3451 

0.1993 

43 

0.0199 

0.0030 

4 

0.2440 

0.1220 

44 

0.0194 

0.0029 

5 

0.1890 

0.0845 

45 

0.0190 

0.0028 

6 

0.1543 

0.0630 

46 

0.0186 

0.0027 

7 

0.1304 

0.0493 

47 

0.0182 

0.0027 

8 

0.1130 

0.0399 

48 

0.0178 

0.0026 

9 

0.0996 

0.0332 

49 

0.0174 

0.0025 

10 

0.0891 

0.0282 

50 

0.0171 

0.0024 

11 

0.0806 

0.0243 

51 

0.0167 

0.0023 

12 

0.0736 

0.0212 

52 

0.0164 

0.0023 

13 

0.0677 

0.0188 

53 

0.0161 

0.0022 

14 

0.0627 

0.0167 

54 

0.0158 

0.0022 

15 

0.0583 

0.0151 

55 

0.0155 

0.0021 

16 

0.0546 

0.0136 

56 

0.0152 

0.0020 

17 

0.0513 

0.0124 

57 

0.0150 

0.0020 

18 

0.0483 

0.0114 

58 

0.0147 

0.0019 

19 

0.0457 

0.0105 

59 

0.0145 

0.0019 

20 

0.0434 

0.0097 

60 

0.0142 

0.0018 

21 

0.0412 

0.0090 

61 

0.0140 

0.0018 

22 

0.0393 

0.0084 

62 

0.0137 

0.0017 

23 

0.0376 

0.0078 

63 

0.0135 

0.0017 

24 

0.0360 

0.0073 

64 

0.0133 

0.0017 

25 

0.0345 

0.0069 

65 

0.0131 

0.0016 

26 

0.0332 

0.0065 

66 

0.0129 

0.0016 

27 

0.0319 

0.0061 

67 

0.0127 

0.0016 

28 

0.0307 

0.0058 

68 

0.0125 

0.0015 

29 

0.0297 

0.0055 

69 

0.0123 

0.0015 

30 

0.0287 

0.0052 

70 

0.0122 

0.0015 

31 

0.0277 

0.0050 

71 

0.0120 

0.0014 

32 

0.0268 

0.0047 

72 

0.0118 

0.0014 

33 

0.0260 

0.0045 

73 

0.0117 

0.0014 

34 

0.0252 

0.0043 

74 

0.0115 

0.0013 

35 

0.0245 

0.0041 

75 

0.0113 

0.0013 

36 

0.0238 

0.0040 

80 

0.0106 

0.0012 

37 

0.0232 

0.0038 

85 

0.0100 

0.0011 

38 

0.0225 

0.0037 

90 

0.0095 

0.0010 

39 

0.0220 

0.0035 

100 

0.0085 

0.0008 

(Merriman,  "Least  Squares.") 


TABLES 


229 


TABLE  XVI.  —  SQUARES  OP  NUMBERS. 


n 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

1.0 

1.000 

1.020 

1.040 

1.061 

1.082 

1.103 

1.124 

1.145 

1.166 

1.188 

22 

1.1 

1.210 

1.232 

1.254 

1.277 

1.300 

1.323 

1.346 

1.369 

1.392 

1.416 

24 

1.2 

1.440 

1.464 

1.488 

1.513 

1.538 

1.563 

1.588 

1.613 

1.638 

1.664 

26 

1.3 

1.690 

1.716 

1.742 

1.769 

1.796 

1.823 

1.850 

1.877 

1.904 

1.932 

28 

1.4 

1.960 

1.988 

2.016 

2.045 

2.074 

2.103 

2.132 

2.161 

2.190 

2.220 

30 

1.5 

2.250 

2.280 

2.310 

2.341 

2.372 

2.403 

2.434 

2.465 

2.496 

2.528 

32 

1.6 

2.560 

2.592 

2.624 

2.657 

2.690 

2.723 

2.756 

2.789 

2.822 

2.856 

34 

1.7 

2.890 

2.924 

2.958 

2.993 

3.028 

3.063 

3.098 

3.133 

3.168 

3.204 

36 

1.8 

3.240 

3.276 

3.312 

3.349 

3.386 

3.423 

3.460 

3.497 

3.534 

3.572 

38 

1.9 

3.610 

3.648 

3.686 

3.725 

3.764 

3.803 

3.842 

3.881 

3.920 

3.960 

40 

2.0 

4.000 

4.040 

4.080 

4.121 

4.162 

4.203 

4.244 

4.285 

4.326 

4.368 

42 

2.1 

4.410 

4.452 

4.494 

4.537 

4.580 

4.623 

4.666 

4.709 

4.752 

4.796 

44 

2.2 

4.840 

4.884 

4.928 

4.973 

5.018 

5.063 

5.108 

5.153 

5.198 

5.244 

46 

23 

5.290 

5.336 

5.382 

5.429 

5.476 

5.523 

5.570 

5.617 

5.664 

5.712 

48 

2.4 

5.760 

5.808 

5.856 

5.905 

5.954 

6.003 

6.052 

6.101 

6.150 

6.200 

50 

25 

6.250 

6.300 

6.350 

6.401 

6.452 

6.503 

6.554 

6.605 

6.656 

6.708 

52 

2.6 

6.760 

6.812 

6.864 

6.917 

6.970 

7.023 

7.076 

7.129 

7.182 

7.236 

54 

27 

7.290 

7.344 

7.398 

7.453 

7.508 

7.563 

7.618 

7.673 

7.728 

7.784 

56 

2.8 

7.840 

7.896 

7.952 

8.009 

8.066 

8.123 

8.180 

8.237 

8.294 

8.352 

58 

2.9 

8.410 

8.468 

8.526 

8.585 

8.644 

8.703 

8.762 

8.821 

8.880 

8.940 

60 

3.0 

9.000 

9.060 

9.120 

9.181 

9.242 

9.303 

9.364 

9.425 

9.486 

9.548 

62 

3.1 

9.610 

9.672 

9.734 

9.797 

9.860 

9.923 

9.986 

10.05 

10.11 

10.18 

6 

3.2 

10.24 

10.30 

10.37 

10.43 

10.50 

10.56 

10.63 

10.69 

10.76 

10.82 

7 

3.3 

10.89 

10.96 

11.02 

11.09 

11.16 

11.22 

11.29 

11.36 

11.42 

11.49 

7 

3.4 

11.56 

11.63 

11.70 

11.76 

11.83 

11.90 

11.97 

12.04 

12.11 

12.18 

7 

3.5 

12.25 

12.32 

12.39 

12.46 

12.53 

12.60 

12.67 

12.74 

12.82 

12.89 

7 

3.6 

12.96 

13.03 

13.10 

13.18 

13.25 

13.32 

13.40 

13.47 

13.54 

14.62 

7 

3.7 

13.69 

13.76 

13.84 

13.91 

13.99 

14.06 

14.14 

14.21 

14.29 

14.36 

8 

3.8 

14.44 

14.52 

14.59 

14.67 

14.75 

14.82 

14.90 

14.98 

15.05 

15.13 

8 

3.9 

15.21 

15.29 

15.37 

15.44 

15.52 

15.60 

15.68 

15.76 

15.84 

15.92 

8 

4.0 

16.00 

16.08 

16.16 

16.24 

16.32 

16.40 

16.48 

16.56 

16.65 

16.73 

8 

4.1 

16.81 

16.89 

16.97 

17.06 

17.14 

17.22 

17.31 

17.39 

17.47 

17.65 

8 

4.2 

17.64 

17.72 

17.81 

17.89 

17.98 

18.06 

18.15 

18.23 

18.32 

18.40 

9 

4.3 

18.49 

18.58 

18.66 

18.75 

18.84 

18.92 

19.01 

19.10 

19.18 

19.27 

9 

4.4 

19.36 

19.45 

19.54 

19.62 

19.71 

19.80 

19.89 

19.98 

20.07 

20.16 

9 

4.5 

20.25 

20.34 

20.43 

20.52 

20.61 

20.70 

20.79 

20.88 

20.98 

21.07 

9 

4.6 

21.16 

21.25 

21.34 

21.44 

21.53 

21.62 

21.72 

21.81 

21.90 

22.00 

9 

4.7 

22.09 

22.18 

22.28 

22.37 

22.47 

22.56 

22.66 

22.75 

22.85 

22.94 

10 

4.8 

23.04 

23.14 

23.23 

23.33 

23.43 

23.52 

23.62 

23.72 

23.81 

23.91 

10 

4.9 

24.01 

24.11 

24.21 

24.30 

24.40 

24.50 

24.60 

24.70 

24.80 

24.90 

10 

5.0 

25.00 

25.10 

25.20 

25.30 

25.40 

25.50 

25.60 

25.70 

25.81 

25.91 

10 

5.1 

26.01 

26.11 

26.21 

26.32 

26.42 

26.52 

26.63 

26.73 

26.83 

26.94 

10 

5.2 

27.04 

27.14 

27.25 

27.35 

27.46 

27.56 

27.67 

27.77 

27.88 

27.98 

11 

5.3 

28.09 

28.20 

28.30 

28.41 

28.52 

28.62 

28.73 

28.84 

28.94 

29.05 

11 

5.4 

29.16 

29.27 

29.38 

29.48 

29.59 

29.70 

29.81 

29.92 

30.03 

30.14 

11 

n 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

(Merriman,  "Least  Squares.") 


230 


THE  THEORY  OF  MEASUREMENTS 


TABLE  XVI. —SQUARES  OF  NUMBERS  (Concluded). 


n 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

5.5 

30.25 

30.36 

30.47 

30.58 

30.69 

30.80 

30.91 

31.02 

31.14 

31.25 

11 

5.6 

31.36 

31.47 

31.58 

31.70 

31.81 

31.92 

32.04 

32.15 

32.26 

32.38 

11 

5.7 

32.49 

32.60 

32  72 

32.83 

32.95 

33.0633.18 

33.29 

33.41 

33.52 

12 

5.8 

33.64 

33.76 

33.87 

33.99 

34.11 

34.2234.34 

34.46 

34.57 

34.69 

12 

5.9 

34.81 

34.93 

35.05 

35.16 

35.28 

35.40 

35.52 

35.64 

35.76 

35.88 

12 

6.0 

36.00 

36.12 

36.24 

36.36 

36.48 

36.60 

36.72 

36.84 

36.97 

37.09 

12 

6.1 

37.21 

37.33 

37.45 

37.58 

37.70 

37.82 

37.95 

38.07 

38.19 

38.32 

12 

6.2 

38.44 

38.56 

38.69 

38.81 

38.94 

39.06 

39.19 

39.31 

39.44 

39.56 

13 

6.3 

39.69 

39.82 

39.94 

40.07 

40.20 

40.32 

40.45 

40.58 

40.70 

40.83 

13 

6.4 

40.96 

41.09 

41.22 

41.34 

41.47 

41.60 

41.73 

41.86 

41.99 

42.12 

13 

6.5 

42.25 

42.38 

42.51 

42.64 

42.77 

42.90 

43.03 

43.16 

43.30 

43.43 

13 

6.6 

43.56 

43.69 

43.82 

43.96 

44.09 

44.22 

44.36 

44.49 

44.62 

44.76 

13 

6.7 

44.89 

45.02 

45.16 

45.29 

45.43 

45.56 

45.70 

45.83 

45.97 

46.10 

14 

6.8 

46.24 

46.38 

46.51 

46.65 

46.79 

46.92 

47.06 

47.20 

47.33 

47.47 

14 

6.9 

47.61 

47.75 

47.89 

48.02 

48.16 

48.30 

48.44 

48.58 

48.72 

48.86 

14 

7.0 

49.00 

49.14 

49.28 

49.42 

49.56 

49.70 

49.84 

49.98 

50.13 

50.27 

14 

7.1 

50.41 

50.55 

50.69 

50.84 

50.98 

51.12 

51.27 

51.41 

51.55 

51.70 

14 

7.2 

51.84 

51.98 

52.13 

52.27 

52.42 

52.56 

52.71 

52.85 

53.00 

53.14 

15 

7.3 

53.29 

53.44 

53.58 

53.73 

53.88 

54.02 

54.17 

54.32 

54.46 

54.61 

15 

7.4 

54.76 

54.91 

55.06 

55.20 

55.35 

55.50 

55.65 

55.80 

55.95 

56.10 

15 

7.5 

56.25 

56.40 

"56.55 

56.70 

56.85 

57.00 

57.15 

57.30 

57.46 

57.61 

15 

7.6 

57.76 

57.91 

58.06 

58.22 

58.37 

58.52 

58.68 

58.83 

58.98 

59.14 

15 

7.7 

59.29 

59.44 

59.60 

59.75 

59.91 

60.06 

60.22 

60.37 

60.53 

60.68 

16 

7.8 

60.84 

61.00 

61.15 

61.31 

61.47 

61.62 

61.78 

61.94 

62.09 

62.25 

16 

7.9 

62.41 

62.57 

62.73 

62.88 

63.04 

63.20 

63.36 

63.52 

63.68 

63.84 

16 

8.0 

64.00 

64.16 

64.32 

64.48 

64.64 

64.80 

64.96 

65.12 

65.29 

65.45 

16 

8.1 

65.61 

65.77 

65.93 

66.10 

66.26 

66.42 

66.59 

66.75 

66.91 

67.08 

16 

8.2 

67.24 

67.40 

67.57 

67.73 

67.90 

68.06 

68.23 

68.39 

68.56 

68.72 

17 

8.3 

68.89 

69.06 

69.22 

69.39 

69.56 

69.72 

69.89 

70.06 

70.22 

70.39 

17 

8.4 

70.56 

70.73 

70.90 

71.06 

71.23 

71.40 

71.57 

71.74 

71.91 

72.08 

17 

8.5 

72.25 

72.42 

72.59 

72.76 

72.93 

73.10 

73.27 

73.44 

73.62 

73.79 

17 

8.6 

73.96 

74.13 

74.30 

74.48 

74.65 

74.82 

75.00 

75.17 

75.34 

75.52 

17 

8.7 

75.69 

75.86 

76.04 

76.21 

76.39 

76.56 

76.74 

76.91 

77.09 

77.26 

18 

8.8 

77.44 

77.62 

77.79 

77.97 

78.15 

78.32 

78.50 

78.68 

78.85 

79.03 

18 

8.9 

79.21 

79.39 

79.57 

79.74 

79.92 

80.10 

80.28 

80.46 

80.64 

80.82 

18 

9.0 

81.00 

81.18 

81.36 

81.54 

81.72 

81.90 

82.08 

82.26 

82.45 

82.63 

18 

9.1 

82.81 

82.99 

83.17 

83.36 

83.54 

83.72 

83.91 

84.09 

84.27 

84.46 

18 

9.2 

84.64 

84.82 

85.01 

85.19 

85.38 

85.56 

85.75 

85.93 

86.12 

86.30 

19 

9.3 

86.49 

86.68 

86.86 

87.05 

87.24 

87.42 

87.61 

87.80 

87.98 

88.17 

19 

9.4 

88.36 

88.55 

88.74 

88.92 

89.11 

89.30 

89.49 

89.68 

89.87 

90.06 

19 

9.5 

90.25 

90.44 

90.63 

90.82 

91.01 

91.20 

91.39 

91.58 

91.78 

91.97 

19 

9.6 

92.16 

92.35 

92.54 

92.74 

92.93 

93.12 

93.32 

93.51 

93.70 

93.90 

19 

9.7 

94.09 

94.28 

94.48 

94.67 

94.87 

95.06 

95.26 

95.45 

95.65 

95.84 

20 

9.8 

96.04 

96.24 

96.43 

96.63 

96.83 

97.02 

97.22 

97.42 

97.61 

97.81 

20 

9.9 

98.01 

98.21 

98.41 

98.60 

98.80 

99.00 

99.20 

99.40 

99.60 

99.80 

20 

n 

0 

l 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

TABLES 
TABLE  XVII.  —  LOGARITHMS;  1000  TO  1409. 


231 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

101 

0043 

0048 

0052 

0056 

0060 

0065 

0069 

0073 

0077 

0082 

102 

0086 

0090 

0095 

0099 

0103 

0107 

0111 

0116 

0120 

0124 

103 

0128 

0133 

0137 

0141 

0145 

0149 

0154 

0158 

0162 

0166 

104 

0170 

0175 

0179 

0183 

0187 

0191 

0195 

0199 

0204 

0208 

105 

0212 

0216 

0220 

0224 

0228 

0233 

0237 

0241 

0245 

0249 

106 

0253 

0257 

0261 

0265 

0269 

0273 

0278 

0282 

0286 

0290 

107 

0294 

0298 

0302 

0306 

0310 

0314 

0318 

0322 

0326 

0330 

108 

0334 

0338 

0342 

0346 

0350 

0354 

0358 

0362 

0366 

0370 

109 

0374 

0378 

0382 

0386 

0390 

0394 

0398 

0402 

0406 

0410 

110 

0414 

0418 

0422 

0426 

0430 

0434 

0438 

0441 

0445 

0449 

111 

0453 

0457 

0461 

0465 

0469 

0473 

0477 

0481 

0484 

0488 

112 

0492 

0496 

0500 

0504 

0508 

0512 

0515 

0519 

0523 

0527 

113 

0531 

0535 

0538 

0542 

0546 

0550 

0554 

0558 

0561 

0565 

114 

0569 

0573 

0577 

0580 

0584 

0588 

0592 

0596 

0599 

0603 

115 

0607 

0611 

0615 

0618 

0622 

0626 

0630 

0633 

0637 

0641 

116 

0645 

0648 

0652 

0656 

0660 

0663 

0667 

0671 

0674 

0678 

117 

0682 

0686 

0689 

0693 

0697 

0700 

0704 

0708 

0711 

0715 

118 

0719 

0722 

0726 

0730 

0734 

0737 

0741 

0745 

0748 

0752 

119 

0755 

0759 

0763 

0766 

0770 

0774 

0777 

0781 

0785 

0788 

120 

0792 

0795 

0799 

0803 

0806 

0810 

0813 

0817 

0821 

0824 

121 

0828 

0831 

0835 

0839 

0842 

0846 

0849 

0853 

0856 

0860 

122 

0864 

0867 

0871 

0874 

0878 

0881 

0885 

0888 

0892 

0896 

123 

0899 

0903 

0906 

0910 

0913 

0917 

0920 

0924 

0927 

0931 

124 

0934 

0938 

0941 

0945 

0948 

0952 

0955 

0959 

0962 

0966 

125 

0969 

0973 

0976 

0980 

0983 

0986 

0990 

0993 

0997 

1000 

126 

1004 

1007 

1011 

1014 

1017 

1021 

1024 

1028 

1031 

1035 

127 

1038 

1041 

1045 

1048 

1052 

1055 

1059 

1062 

1065 

1069 

128 

1072 

1075 

1079 

1082 

1086 

1089 

1092 

1096 

1099 

1103 

129 

1106 

1109 

1113 

1116 

1119 

1123 

1126 

1129 

1133 

1136 

130 

1139 

1143 

1146 

1149 

1153 

1156 

1159 

1163 

1166 

1169 

131 

1173 

1176 

1179 

1183 

1186 

1189 

1193 

1196 

1199 

1202 

132 

1206 

1209 

1212 

1216 

1219 

1222 

1225 

1229 

1232 

1235 

133 

1239 

1242 

1245 

1248 

1252 

1255 

1258 

1261 

1265 

1268 

134 

1271 

1274 

1278 

1281 

1284 

1287 

1290 

1294 

1297 

1300 

135 

1303 

1307 

1310 

1313 

1316 

1319 

1323 

1326 

1329 

1332 

136 

1335 

1339 

1342 

1345 

1348 

1351 

1355 

1358 

1361 

1364 

137 

1367 

1370 

1374 

1377 

1380 

1383 

1386 

1389 

1392 

1396 

138 

1399 

1402 

1405 

1408 

1411 

1414 

1418 

1421 

1424 

1427 

139 

1430 

1433 

1436 

1440 

1443 

1446 

1449 

1452 

1455 

1458 

140 

1461 

1464 

1467 

1471 

1474 

1477 

1480 

1483 

1486 

1489 

(Bottomley,  "Four  Fig.  Math.  Tables.") 


232 


THE   THEORY  OF  MEASUREMENTS 


*  TABLE  XVIII.  —  LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

& 

9 

123 

456 

789 

10 

0000 

0043 

0086 

0128 

0170 

O2I2 

0253 

0294 

0334 

0374 

4812 

17  21  25 

29  33  37 

11 

12 
13 

0414 
0792 

"39 

0453 
0828 

"73 

0492 
0864 
1206 

0531 
0899 
1239 

0569 

0934 
1271 

0607 
0969 
I3°3 

0645 
100^ 

1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1106 

1430 

4811 
3  7io 
3  6  10 

15  19  23 
14  17  21 

13  16  10 

26  30  34 
24  28  31 
23  26  29 

21  24  27 
20  22  25 

18  21  24 

14 
15 
16 

1461 
1761 
2041 

1492 
1790 
2068 

iffi 

2095 

1553 

1847 

2122 

1584 
1875 
2148 

i6i<: 
1903 
2175 

164^: 

1931 

22OI 

1673 
1959 
2227 

1703 
1987 

2253 

1732 
2014 
2279 

3  6  9 
36  8 

3  5  8 

12  15  18 
ii  14  17 
ii  13  16 

17 
18 
19 

2304 

$1 

2330 
2577 
2810 

2355 
2601 

2833 

2380 
2625 
2856 

2405 
2648 
2878 

2430 
2672 
2900 

2455 
2695 

2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

257 

2  5  7 
247 

10  12  15 

9  12  14 
9  "  I2 

17  2O  22 

16  19  21 
16  18  20 

20 

3010 

3032 

3054 

3°75 

3096 

3"8 

3139 

3160 

3181 

3201 

24  6 

8  ii  13 

15  17  19 

21 
22 
23 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 

37" 

3345 
354i 
3729 

3365 
3560 

3747 

3385 
3579 
3766 

3404 
3598 
3784 

2  4  6 
24  6 
2  4  6 

8  10  12 
8  10  12 

7  9  ii 

14  16  18 
H  15  17 
J3  15  '7 

24 
25 
26 

3802 
3979 
415° 

3820 

3997 
4166 

3838 
4014 
4183 

3856 

4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 

4133 
4298 

245 

235 
235 

7  9  ii 
7  9  10 
7  8  10 

12  14  16 

12  14  15 
II  13  15 

27 
28 
29 

43H 
4472 
4624 

4330 
4487 

4639 

4346 
4502 

4654 

4362 
45l8 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
47J3 

4425 
4579 
4728 

444° 
4594 
4742 

4456 
4609 

4757 

2  3  5 

2  3  5 
1  3  4 

689 
689 
6  7  9 

II  13  14 
II  12  \i 
10  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

i  3  4 

6  7  9 

10  ii  13 

31 
32 
33 

4914 

505i 
5185 

4928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 

5224 

4969 
5105 
5237 

4983 
5"9 
5250 

4997 
5i|2 
5263 

5011 

5*45 
5276 

5024 

5159 
5289 

5038 
5172 
5302 

3  4 
3  4 
3  4 

6  7  8 

HI 

10  II  12 
9  II  12 
9  10  12 

34 
35 
36 

5315 
544i 
5563 

5328 
5453 
5575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
55»4 
5635 

5403 
5527 
5647 

54i6 

5539 
5658 

5428 

555i 
5670 

3  4 
2  4 
2  4 

!.:; 

5  6  7 

9  10  ii 
9  10  ii 
8  10  ii 

37 
38 
39 

5682 
5798 
59" 

5694 
5809 
5922 

5705 
5821 

5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 

58II 
5966 

5977 

5763 
5877 
5988 

577C 
5999 

5786 

5899 
6010 

2  3 
2  3 
2  3 

5  6  7 

5  6  7 
4  5  7 

8  9  10 
8  9  10 
8  9  10 

40 

6021 

6031 

6042 

6o53 

6064 

6075 

6085 

6096 

6107 

6117 

2  3 

4  5  6 

8  9  10 

41 
42 
43 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 

6314 
6415 

6222 
6325 
6425 

2  3 
2  3 

2  3 

4  5  6 
4  5  6 
4  5  6 

7  8  9 
7  8  9 
7  8  9 

44 
45 
46 

6435 
6532 
6628 

6444 
6542 
6637 

6454 
^6 

6464 
6561 
6656 

6474 

6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

65°3 
6599 

6513 
6609 
6702 

6522 
6618 
6712 

2  3 

I  2  3 

I  2  3 

4  5  6 

4  5  6 
456 

7  8  9 
7  8  9 
7  7  8 

47 
48 
49 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
683O 
6920 

6749 
6839 
6928 

6758 
6848 

6937 

6767 
6857 
6946 

6776 
6866 
6955 

6785 

?25 
6964 

6794 
6884 
6972 

6803 
6893 
6981 

I  2  3 
I  2  3 
I  2  3 

4  5  5 
4  4  5 
445 

6  7  8 
678 
678 

50 

6990 

6998 

7007 

7016 

7024 

7°33 

7042 

7050 

7059 

7067 

I  2  3 

3  4  5 

678 

51 
52 
53 

7076 
7160 
7243 

7084 
7168 
7251 

7093 
7177 

7259 

7101 

7185 
7267 

7110 
7193 
7275 

7118 
7202 
7284 

7126 
7210 
7292 

7i35 
7218 
7300 

7H3 
7226 
7308 

7152 
7235 
73i6 

I  2  3 
122 
I  2  2 

3  4  5 
3  4  5 
345 

678 

6  7  7 
667 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

I  2  2 

3  4  5 

667 

*  From  Bottomley's  Four  Figure  Mathematical  Tables,  by  courtesy  of  The  Macmillan  Company. 


TABLES 


233 


TABLE  XVIII.  —  LOGARITHMS  (Concluded). 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1  23 

456 

789 

55 

7404 

.7412 

7490 
£566 
^642 

74i9 

7427 

7435 

7443 

745i 

7459 

7466 

7474 

122 

3  4  5 

5  6  7 

56 
57 
58 

7482 
7559 
7634 

7497 
7574 
7649 

7505 

7582 

7657 

75*3 
7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

7536 
7612 
7686 

7543 
7619 
7694 

755i 
7627 
7701 

2  2 
2  2 
I   2 

345 
3  4  5 
344 

5  6  7 
5  6  7 
5  6  7 

59 
60 
61 

7709 
7782 
7853 

7716 

7789 
7860 

7723 
7796 
7868 

773i 
7803 
7875 

7738 
7810 
7882 

7745 
7818 
7889 

Ws 

7896 

7760 
7832 
7903 

7767 

7839 
7910 

7774 
7846 
7917 

2 
2 
2 

344 
344 
344 

5  6  7 
566 
5  6  6 

62 
63 
64 

7924 

7993 
8062 

7931 
8000 
8069 

7938 
8007 

8075 

7945 
8014 
8082 

7952 
8021 
8089 

79^Q 

8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 
8048 
8116 

7987 
8055 
8122 

2 
2 
2 

334 
334 
334 

566 
5  5  6 
5  5  6 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

2 

334 

5  5  6 

66 
67 
68 

.8195 
8261 

8325 

8202 
8267 
833i 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 

8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 

8319 
8382 

2 

2 
2 

334 
334 
334 

5  5  6 

5  5  \ 
4  5  6 

69 
70 
71 

8388 
8451 
8513 

8395 
8457 
8519 

8401 
8463 
8525 

8407 
8470 
8531 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 
8555 

8439 
8500 
8561 

8445 
8506 
8567 

2 
2 
2 

234 
234 
234 

4  5  6 
4  5  6 
4  5  5 

72 

73 

74 

~75~ 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 
8745 

2 
2 

2 

234 
234 
234 

455 
455 
4  5  5 

875i 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

2 

233 

4  5  5 

76 
77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

8859 

8915 
8971 

2 
2 
2 

233 
233 
233 

4  5  5 
4  4  5 
4  4  5 

445 
4  4  5 
445 

79 
80 
81 

8976 
9031 
9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 

9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 

9025 
9079 
9U3 

2 

2 
2 

233 
233 
233 

82 
83 
84 

9138 
9191 

9243 

9H3 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 

9263 

9165 

9217 
9269 

9170 
9222 
9274 

9175 
9227 

9279 

9180 
9232 
9284 

9186 
9238 
9289 

2 
2 
2 

233 
233 
233 

4  4  5 
445 
445 

85 

9294 

9299 

9304 

9309 

93i5 

9320 

9325 

9330 

9335 

9340 

I     2 

233 

445 

86 
87 
88 

9345 
9395 
9445 

935° 
9400 

945° 

9355 
9405 
9455 

9360 
9410 
9460 

9365 
94i5 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

938o 
943° 
9479 

9385 
9435 
9484 

9390 
9440 
9489 

I     2 
O 
0 

233 
223 
223 

4  4  5 
344 
344 

89 
90 
91 

9494 
9542 
9590 

9499 
9547 
9595 

95°4 
9552 
9600 

95°9 
9557 
9605 

95*3 
9562 
9609 

9518 
9566 
9614 

9523 
957i 
9619 

9528 
9576 
9624 

9533 
9628 

9538 
9586 

9633 

O 
0 
O 

223 
223 
223 

344 
344 
344 

92 
93 
94 

~95~ 

9638 
9685 
973i 

9643 
9689 

9736 

9647 
9694 
974i 

9652 
9699 
9745 

9657 
97°3 
975° 

9661 
9708 

9754 

9666 

97  i  3 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

0 
O 
O 

223 
223 
223 

344 
344 
344 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

0 

223 

344 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 
9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

9845 
9890 

9934 

9850 
9894 
9939 

9854 
9899 
9943 

9859 
9903 
9948 

9863 
9908 
9952 

O 
O 
O 

223 
223 
223 

344 
344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

0  I   I 

223 

334 

234 


THE   THEORY  OF  MEASUREMENTS 


*  TABLE  XIX.  —  NATURAL  SINES. 


0' 

6' 

12' 

18' 

24' 

SO' 

36' 

42' 

48' 

54' 

123 

4  5 

0° 

oooo 

0017 

oo35 

0052 

0070 

0087 

0105 

OI22 

0140 

oi57 

369 

12  15 

1 

2 
3 

0175 
0349 
0523 

0192 

0366 
0541 

0209 
0384 
0558 

0227 
0401 
0576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

0297 
0471 
0645 

0314 
0488 
o663 

0332 
0506 
0680 

369 
369 
369 

12   I5 
12   I5 
12   I5 

4 
5 
6 

~7~ 
8 
9 

0698 
0872 
1045 

°7!5 
0889 
1063 

0732 
0906 
1080 

0750 
0924 
1097 

0767 
0941 
"15 

0785 
0958 
1132 

0802 
0976 
1149 

0819 

0993 
1167 

0837 
ion 
1184 

0854 
1028 

I2OI 

369 
369 
369 

12   I5 
12   14 

12   14 

1219 

1392 
1564 

1236 
1409 
1582 

1253 
1426 

1599 

1271 

1444 
1616 

1288 
1461 
1633 

1305 
1478 
1650 

J323 

1495 
1668 

1340 

\&1 

1357 
1530 
1702 

!374 
J547 
1719 

369 
369 
369 

12   14 
12   14 
12   14 

10 

1736 

!754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

369 

12   14 

11 
12 
13 

1908 
2079 

2250 

1925 
2096 
2267 

1942 
2113 

2284 

1959 
2130 
2300 

1977 
2147 
2317 

1994 
2164 
2334 

2OII 

2181 

235  I 

2028 
2198 
2368 

2045 
2215 
2385 

2062 
2232 
2402 

369 
369 
368 

II   I4 
II   14 

II   I4 

14 
15 
16 

TT 
18 
19 

2419 
2588 
2756 

2436 
2605 
2773 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 
2689 

2857 

2538 
2706 
2874 

2554 
2723 
2890 

257i 
2740 
2907 

368 
368 
368 

II   14 
II   I4 
II   14 

2924 
3090 
3256 

2940 
3io7 
3272 

2957 
3123 
3289 

2974 
3J4° 
3305 

2990 
3156 
3322 

3007 
3i73 
3338 

3024 
3190 

3355 

3040 
3206 

3371 

3057 
3223 
3387 

3074 
3239 
3404 

3  6  8 
368 

3  5  8 

II   14 
II   14 
II   14 

20 

3420 

3437 

3453 

3469 

3486 

3502 

35i8 

3535 

3551 

3567 

3  5  8 

II   14 

21 
22 
23 

~24~ 
25 
26 

3584 
3746 
3907 

3600 
3762 
3923 

3616 
3778 
3939 

3633 
3795 
3955 

3649 
3811 

3971 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
4019 

37H 
3875 
4035 

3730 
3891 
405  l 

3  5  8 
3  5  8 
3  5  8 

II   14 
II   14 
II   14 

4067 

4226 

4384 

4083 
4242 
4399 

4099 
4258 
4415 

4H5 

4274 
443i 

4131 

4289 
4446 

4147 

43°5 
4462 

4163 
432i 
4478 

4179 
4337 
4493 

4195 
4352 
45°9 

4210 
4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II   13 
II   13 

10  13 

27 
28 
29 

4540 

4695 
4848 

4555 
4710 
4863 

457  i 
4726 
4879 

4586 
474i 
4894 

4602 

4756 
4909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 
4818 
497° 

4679 
4833 
4985 

3  5  8 
3  5  8 

3  5  8 

10  13 

10  13 
10  13 

30 

5000 

5015 

5°3o 

5°45 

5060 

5075 

5090 

5I05 

5120 

5135 

3  5  8 

10  13 

31 
32 
33 

5150 
5299 
5446 

5*65 
53H 
546i 

5180 
5329 
5476 

5195 
5344 
5490 

5210 
5358 
5505 

5225 
5373 
5519 

5240 
5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5284 
5432 
5577 

2  5  7 
257 
2  5  7 

IO   12 
10   12 
IO   12 

34 
35 
36 

5592 
5736 
5878 

5606 

575° 
5892 

5621 

5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693 
5835 
5976 

57°7 
5850 
5990 

572i 
5864 
6004 

257 

2  5  7 
2  5  7 

IO   12 
IO   12 

9  12 

37 
38 
39 

6018 

6157 
6293 

6032 
6170 
6307 

6046 
6184 
6320 

6060 
6198 
6334 

6074 
6211 
6347 

6088 
6225 
6361 

6101 
6239 
6374 

6115 
6252 
6388 

6129 
6266 
6401 

6143 
6280 
6414 

257 
2  5  7 
247 

9  12 
9  ii 
9  ii 

40 

6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

247 

9  ii 

41 
42 
43 

6561 
6820 

6574 
6704 

6833 

6587 
6717 
6845 

6600 
6730 
6858 

6613 

6743 
6871 

6626 
6756 
6884 

6639 
6769 
6896 

6652 
6782 
6909 

6665 
6794 
6921 

6678 
6807 
6934 

247 
2  4  6 
246 

9  ii 

9  " 
8  ii 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

246 

8  10 

*  From  Bottomley's  Four  Figure  Mathematical  Tables,  by  courtesy  of  The  Macmillan  Company. 


TABLES 
TABLE  XIX.  —  NATURAL  SINES  (Concluded). 


235 


0' 

6' 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

45° 

7071 

7083 

7096 

7108 

7120 

7J33 

7H5 

7i57 

7169 

7181 

246 

8  10 

46 

47 
48 

7J93 
73*4 
743i 

7206 
7325 

7443 

7218 
7337 

7455 

7230 

7349 
7466 

7242 
7361 

7478 

7254 
7373 
7490 

7266 

7385 
7501 

7278 
7396 
75i3 

7290 
7408 
7524 

7302 
7420 
7536 

246 
246 
246 

8  10 
8  10 
8  10 

49 
50 
51 

7547 
7660 
7771 

7558 
7672 
7782 

757° 
7683 
7793 

758i 
7694 
7804 

7593 
7705 
7815 

7604 
7716 
7826 

76l5 

7727 
7837 

7627 
7738 
7848 

7638 
7749 
7859 

7649 
7760 
7869 

2  4  6 
246 
2  4  5 

8   9 
7   9 
7   9 

52 
53 
54 

7880 
7986 
8090 

7891 

7997 
8100 

7902 
8007 
8111 

7912 
8018 
8121 

7923 
8028 
8131 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

7965 
8070 
8171 

7976 
8080 
8181 

2  4  5 
235 
2  3  5 

7   9 
7   9 
7   8 

55 

8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

2  3  5 

7   8 

56 
57 
58 

8290 

8387 
8480 

8300 
8396 
8490 

8310 
8406 
8499 

8320 

8415 
8508 

8329 
8425 
8517 

8339 
8434 
8526 

8348 
8443 
8536 

8358 
8453 
8545 

8368 
8462 
8554 

8377 
8471 

8563 

2  3  5 
2  3  5 
2  3  5 

6   8 
6   8 
6   8 

59 
60 
61 

8572 
8660 
8746 

8581 
8669 
8755 

8590 
8678 
8763 

8599 
8686 
8771 

8607 
8695 
8780 

8616 
8704 
8788 

8625 
8712 
8796 

8634 
8721 
8805 

8643 
8729 
8813 

8652 
8738 
8821 

i  3  4 
i  3  4 
i  3  4 

6   7 
2   ? 

62 
63 
64 

8829 
8910 
8988 

8838 
8918 
8996 

8846 
8926 
9003 

8854 
8934 
9011 

8862 
8942 
9018 

8870 
8949 
9026 

8878 
8957 
9033 

8886 
8965 
9041 

8894 

8973 
9048 

8902 
8980 
9056 

i  3  4 
i  3  4 
i  3  4 

1   I 

5   6 

65 

9063 

9070 

9078 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

I   2   4 

5   6 

66 
67 
68 

9135 
9205 
9272 

9M3 
9212 
9278 

915° 
9219 
9285 

9157 
9225 
9291 

9164 
9232 
9298 

9171 
9239 
93°4 

9178 
9245 
93" 

9184 
9252 
9317 

9191 
9259 
9323 

9198 
9265 
9330 

I   2   3 
I   2   3 
I   2   3 

5   6 
4   6 

4   5 

69 
70 
71 

9336 
9397 
9455 

9342 
9403 
9461 

9348 
9409 
9466 

9354 
94i5 
9472 

936i 
9421 
9478 

9367 
9426 

9483 

9373 
9432 
9489 

9379 
9438 
9494 

9385 
9444 
9500 

939i 
9449 
95°5 

2  3 
2  3 
2  3 

4   5 
4   5 
4   5 

72 
73 

74 

9511 
9563 
9613 

95l6 
9568 
9617 

952i 
9573 
9622 

9527 
9578 
9627 

9532 
9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

9548 
9598 
9646 

9553 
9603 
9650 

9558 
9608 

9655 

2  3 

2   2 
2   2 

4   4 
3   4 
3   4 

75 

9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

I   2 

3   4 

76 

77 
78 

9703 
9744 
9781 

9707 
9748 
9785 

9711 
975i 
9789 

9715 

9755 
9792 

9720 

9759 
9796 

9724 
9763 
9799 

9728 
9767 
9803 

9732 

977° 
9806 

9736 
9774 
9810 

9740 
9778 
9813 

2 
2 
2 

3   3 
3   3 
2   3 

79 
80 
81 

9816 
9848 
9877 

9820 

9851 
9880 

9823 
9854 
9882 

9826 

9857 
9885 

9829 
9860 
9888 

9833 
9863 
9890 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I      2 

O 
O 

2   3 

2    2 
2    2 

82 
83 
84 

9903 
9925 
9945 

9905 
9928 

9947 

9907 
9930 
9949 

9910 
9932 
995i 

9912 
9934 
995  2 

9914 
9936 
9954 

9917 
9938 
9956 

9919 
9940 
9957 

992i 
9942 
9959 

9923 
9943 
9960 

O 
O 
0 

2    2 
I    2 
I    I 

85 

9962 

9963 

9965 

9966 

9968 

9969 

9971 

9972 

9973 

9974 

001 

I    I 

86 
87 
88 

9976 
9986 
9994 

9977 
9987 
9995 

9978 
9988 

9995 

9979 
9989 
9996 

9980 
9990 
9996 

998i 
9990 

9997 

9982 
9991 
9997 

9983 
9992 
9997 

9984 
9993 
9998 

9985 
9993 
9998 

0   0   I 
O   O   O 
O   O   O 

I    I 
I    I 

O    O 

89 

9998 

9999 

9999 

9999 

9999 

I'OOO 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

I'OOO 
nearly. 

I'OOO 

nearly. 

O   O   O 

O    O 

236 


THE   THEORY  OF  MEASUREMENTS 


*  TABLE  XX.  —  NATURAL  COSINES. 


O' 

& 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

0° 

I  '000 

I'OOO 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

rooo 

nearly. 

9999 

9999 

9999 

9999 

9999 

o  o  o 

0    0 

1 

2 
3 

9998 

9994 
9986 

9998 

9998 

9993 
9984 

9997 
9992 
9983 

9997 
9991 
9982 

9997 
9990 
9981 

9996 
9990 
9980 

9996 
9989 
9979 

9995 
9988 

9978 

9995 
9987 
9977 

000 

o  o  o 

O  O  I 

0    0 

I   I 
I   I 

4 
5 
6 

9976 
9962 
9945 

9974 
9960 

9943 

9973 
9959 
9942 

9972 
9957 
9940 

9971 

9956 
9938 

9969 
9954 
9936 

9968 
9952 
9934 

9966 

9951 
9932 

9965 
9949 
9930 

9963 
9947 
9928 

o  o 

0   I 
O   I 

I   I 

I    2 
I    2 

7 
8 
9 

9925 
9903 
9877 

9923 
9900 

9874 

9921 
9898 
9871 

9919 

9895 
9869 

9917 

9893 
9866 

9914 
9890 
9863 

9912 
9888 
9860 

9910 
9885 
9857 

9907 
9882 
9854 

9905 
9880 

9851 

0   I 
O   I 
0   I   I 

2    2 
2    2 
2    2 

10 

9848 

9845 

9842 

9839 

9836 

9833 

9829 

9826 

9823 

9820 

112 

2    3 

11 
12 
13 

9816 
9781 
9744 

9813 
9778 
9740 

9810 
9774 
9736 

9806 

977° 
9732 

9803 
9767 
9728 

9799 
9763 
9724 

9796 

9759 
9720 

9792 
9755 
9715 

9789 

9751 
9711 

9785 
9748 
9707 

112 
I  I  2 
I  I  2 

2    3 

3   3 
3   3 

14 
15 
16 

97°3 
9659 
9613 

9699 

9655 
9608 

9694 
9650 
9603 

9690 
9646 
9598 

9686 
9641 
9593 

9681 
9636 
9588 

9677 
9632 
9583 

9673 
9627 

9578 

9668 
9622 
9573 

9664 
9617 
9568 

I  I  2 
122 
122 

3   4 
3   4 
3   4 

17 
18 
19 

9563 
9511 

9455 

9558 
95°5 
9449 

9553 
9500 

9444 

9548 
9494 
9438 

9542 
9489 
9432 

9537 
9483 
9426 

9532 
9478 
9421 

9527 
9472 

94i5 

9521 
9466 
9409 

95i6 
9461 
9403 

I  2  3 

i  2  3 

I  2  3 

4   4 
4   5 
4   5 

20 

9397 

939i 

9385 

9379 

9373 

9367 

9361 

9354 

9348 

9342 

I  2  3 

4   5 

21 
22 
23 

9336 
9272 
9205 

9330 
9265 
9198 

9323 
9259 
9191 

9317 
9252 
9184 

93" 
9245 
9178 

93°4 
9239 
9171 

9298 
9232 
9164 

9291 
9225 
9157 

9285 
9219 
915° 

9278 
9212 
9H3 

I  2  3 
I  2  3 
I  2  3 

4   5 
4   6 
5   6 

24 
25 
26 

9135 
9063 
8988 

9128 
9056 
8980 

9121 
9048 
8973 

9114 
9041 
8965 

9107 
9033 
8957 

9100 
9026 
8949 

9092 
9018 
8942 

9085 
9011 
8934 

9078 
9003 
8926 

9070 
8996 
8918 

I  2  4 

i  3  4 
i  3  4 

5   6 
5   6 
5   6 

27 
28 
29 

8910 
8829 
8746 

8902 
8821 
8738 

8894 
8813 
8729 

8886 
8805 
8721 

8878 
8796 
8712 

8870 
8788 
8704 

8862 
8780 
8695 

8854 
8771 
8686 

8846 

8763 
8678 

8838 

8755 
8669 

i  3  4 
i  3  4 
i  3  4 

5   7 
6   7 
6   7 

30 

8660 

8652 

8643 

8634 

8625 

8616 

8607 

8599 

8590 

8581 

1  3  4 

6   7 

31 
32 
33 

8572 
8480 

8387 

8563 
8471 
8377 

8462 
8368 

8545 
8453 
8358 

8536 
8443 
8348 

8526 
8434 
8339 

8517 
8425 
8329 

8508 

8415 
8320 

8499 
8406 
8310 

8490 
8396 
8300 

2  3  5 
2  3  5 
235 

6   8 

6   8 
6   8 

34 
35 
36 

8290 
8192 
8090 

8281 
8181 
8080 

8271 
8171 
8070 

8261 
8161 
8059 

8251 
8151 
8049 

8241 
8141 
8039 

8231 
8131 
8028 

8221 
8121 
8018 

8211 
8111 
8007 

8202 
8100 
7997 

2  3  5 
2  3  5 
235 

7   8 
7   8 
7   9 

37 
38 
39 

7986 
7880 
7771 

7976 
7869 
7760 

7965 
7859 
7749 

7955 
7848 
7738 

7944 
7837 
7727 

7934 
7826 
7716 

7923 
78i5 
77°5 

7912 
7804 
7694 

7902 

7793 
7683 

7891 
7782 
7672 

245 
245 
246 

7   9 
7   9 
7   9 

40 

7660 

7649 

7638 

7627 

76l5 

7604 

7593 

758i 

757° 

7559 

2  4  6 

8   9 

41 
42 
43 

7547 
7431 
73H 

7536 
7420 
7302 

7524 
7408 
7290 

7513 
7396 
7278 

7501 

73fl 
7266 

7490 
7373 
7254 

7478 
736i 
7242 

7466 
7349 
7230 

7455 
7337 
7218 

7443 
7325 
7206 

246 
246 
2  4  6 

8  10 
8  10 
8  10 

44 

7'93 

7181 

7169 

7157 

7H5 

7133 

7120 

7108 

7096 

7083 

2  4  6 

8  10 

N.B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 
*  From  Bottomley'g  Four  Figure  Mathematical  Tables,  by  courtesy  of  The  Macmillan  Company. 


TABLES 


237 


TABLE  XX.  —  NATURAL  COSINES  (Concluded). 


O' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

123 

4  5 

45° 

7071 

7°59 

7046 

7°34 

7022 

7009 

6997 

6984 

6972 

6959 

246 

8  10 

46 

47 
48 

6947 
6820 
6691 

6934 
6807 
6678 

6921 
6794 
6665 

6909 
6782 
6652 

6896 

6769 
6639 

6884 
6756 
6626 

6871 

6743 
6613 

6858 
6730 
6600 

6845 
6717 
6587 

6833 
6704 

6574 

246 
2  4  6 

247 

8  ii 
9  u 
9  ii 

49 
50 
51 

6561 
6428 
6293 

6547 
6414 
6280 

6534 
6401 
6266 

6521 
6388 
6252 

6508 

6374 
6239 

6494 
6361 
6225 

6481 

6347 
6211 

6468 

6334 
6198 

6455 
6320 
6184 

6441 
6307 
6170 

247 
247 
2  5  7 

9  ii 
9  ii 
9  ii 

52 
53 
54 

6l57 
6018 
5878 

6i43 
6004 
5864 

6129 
5990 
5850 

6115 
5976 
5835 

6101 

5962 
5821 

6088 
5948 
58-07 

6074 
5934 
5793 

6060 
5920 
5779 

6046 
5906 
5764 

6032 
5892 
5750 

2  5  7 
257 
257 

9  12 
9  12 
9  12 

55 

5736 

572i 

5707 

5693 

5678 

5664 

5650 

5635 

5621 

5606 

2  5  7 

10   12 

56 
57 
58 

5592 
5446 
5299 

5577 
5432 
5284 

5563 
54i7 
5270 

5548 
5402 

5255 

5534 
5388 
5240 

55i9 
5373 
5225 

5505 
5358 
5210 

5490 
5344 
5J95 

5476 
5329 
5180 

546i 
53H 
5l65 

2  5  7 
2  5  7 
257 

10   12 
10   12 
10   12 

59 
60 
61 

5Z5° 
5000 
4848 

5i35 
4985 
4833 

5120 
4970 
4818 

5105 

4955 
4802 

5090 
4939 
4787 

5°75 
4924 
4772 

5060 
4909 
4756 

5045 
4894 
474i 

5030 
4879 
4726 

5°i5 
4863 
4710 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 
10  13 

62 
63 
64 

4695 
4540 
4384 

4679 

4524 
4368 

4664 
45°9 
4352 

4648 
4493 
4337 

4633 
4478 

4321 

4617 
4462 
4305 

4602 
4446 
4289 

4586 
443i 
4274 

4571 
4415 
4258 

4555 
4399 
4242 

3  5  8 
3  5  8 
3  5  8 

10  13 
10  13 

II  13 

65 

4226 

4210 

4195 

4179 

4163 

4J47 

4131 

4"5 

4099 

4083 

3  5  8 

II  13 

66 
67 
68 

4067 
3907 
3746 

405  I 
3891 
3730 

4035 
3875 
37H 

4019 
3859 
3697 

4003 

3843 
3681 

3987 
3827 
3665 

3971 
3811 

3649 

3955 
3795 
3633 

3939 
3778 
3616 

3923 
3762 
3600 

3  5  8 
3  5  8 
3  5  8 

II  14 
II  14 
II  14 

69 
70 

71 

3584 
3420 
3256 

3567 
3404 
3239 

355i 
3387 
3223 

3535 
337i 
3206 

35i8 

3355 
3190 

3502 
3338 
3173 

3486 
3322 
3156 

3469 
3305 
3140 

3453 
3289 
3123 

3437 
3272 
3J07 

3  5  8 
3  5  8 
3  6  8 

II  14 
II  14 
II  14 

72 
73 

74 

3090 
2924 
2756 

3°74 
2907 
2740 

3057 
2890 
2723 

3040 
2874 
2706 

3024 

2857 
2689 

3007 
2840 
2672 

2990 
2823 
2656 

2974 
2807 
2639 

2957 
2790 
2622 

2940 

2773 
2605 

368 
368 
368 

II  14 

II  14 

II  14 

75 

2588 

257i 

2554 

2538 

2521 

2504 

2487 

2470 

2453 

2436 

368 

II  14 

76 

77 
78 

2419 
2250 
2079 

2402 
2233 
2062 

2385 
2215 
2045 

2368 
2198 
2028 

2351 
2181 

2OII 

2334 
2164 
1994 

2317 
2147 
1977 

2300 
2130 
1959 

2284 
2113 
1942 

2267 
2096 
1925 

368 
369 
369 

II  14 
II  14 

II  14 

79 
80 
81 

1908 
1736 
i564 

1891 
1719 

'547 

1874 
1702 
1530 

1857 
1685 
1513 

1840 
1668 
1495 

1822 
1650 
1478 

1805 

1633 
1461 

1788 
1616 

1444 

1771 

1599 
1426 

*754 
1582 
1409 

369 
369 
369 

12   14 
12   14 
12   14 

82 
83 
84 

1392 
1219 
1045 

1374 

I2OI 
1028 

1357 
1184 

IOII 

1340 
1167 

0993 

1323 
1149 
0976 

1305 
1132 
0958 

1288 

i"5 

0941 

1271 
1097 
0924 

1253 
1080 
0906 

1236 
1063 
0889 

369 
369 
369 

12   I4 
12   I4 
12   14 

85 

0872 

0854 

0837 

0819 

0802 

0785 

0767 

0750 

0732 

0715 

369 

12   I5 

86 
87 
88 

0698 
0523 
0349 

0680 
0506 
0332 

o663 
0488 
03H 

0645 
0471 
0297 

0628 

0454 
0279 

0610 
0436 
0262 

0593 
0419 
0244 

0576 
0401 
0227 

0558 
0384 
0209 

0541 
0366 
0192 

369 
369 
369 

12   15 
12   15 
12   15 

89 

oi75 

0157 

0140 

0122 

0105 

0087 

0070 

0052 

0035 

0017 

369 

12   15 

iV.B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 


238 


THE   THEORY  OF  MEASUREMENTS 


TABLE  XXI.  —  NATURAL  TANGENTS. 


O' 

& 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

0° 

•oooo 

0017 

0035 

0052 

0070 

0087 

0105 

OI22 

0140 

oi57 

369 

12  14 

1 

2 
3 

•0175 

•0349 
•0524 

0192 
0367 
0542 

0209 
0384 
0559 

0227 
0402 
°577 

0244 
0419 
0594 

0262 

0437 
0612 

0279 

0454 
0629 

0297 
0472 
0647 

0314 

0489 
0664 

0332 
0507 
0682 

369 
369 
369 

12  I5 
12  15 
12  I5 

4 
5 
6 

•0699 
•0875 

•1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 
"39 

0805 
0981 
"57 

0822 
0998 

"75 

0840 
1016 
1192 

0857 
1033 

I2IO 

369 
369 
369 

12  I5 
12  I5 
12  I5 

7 
8 
9 

•1228 

•1405 
•1584 

1246 

1423 
1602 

1263 
1441 
1620 

1281 

H59 
1638 

1299 

H77 
1655 

1317 

H95 
1673 

1334 
1512 
1691 

1352 
1530 
1709 

1370 
1548 
1727 

1388 
1566 
1745 

369 
369 
369 

12  I5 
12  I5 
12  I5 

10 

•1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

369 

12  I5 

11 
12 
13 

•1944 

•2126 

•2309 

1962 

2144 
2327 

1980 
2162 

2345 

1998 
2180 
2364 

2016 
2199 

2382 

2035 
2217 
2401 

2053 

2235 
2419 

2071 

2254 
2438 

2089 
2272 
2456 

2IO7 
2290 
2475 

369 
369 
369 

12  I5 

12  I5 
12  I5 

14 
15 
16 

•2493 
•2679 
•2867 

2512 

2698 
2886 

253° 

2717 
2905 

2549 
2736 

2924 

2568 
2754 
2943 

2586 

2773 
2962 

2605 
2792 
2981 

2623 
2811 
3000 

2642 
2830 
3019 

2661 
2849 
3038 

369 
369 
369 

12  l6 

13  16 
13  16 

17 
18 
19 

•3057 
•3249 
•3443 

3076 
3269 
3463 

3096 
3288 
3482 

3"5 

3307 
3502 

3134 
3327 
3522 

3i53 
3346 
354i 

3172 
3365 
356i 

3J9i 

3385 
358i 

3211 

3404 
3600 

3230 
3424 
3620 

3  6  10 
3  6  10 
3  6  10 

13  16 
13  16 
13  17 

20 

•3640 

3659 

3679 

3699 

37J9 

3739 

3759 

3779 

3799 

3819 

3  7  I0 

13  17 

21 
22 
23 

•3839 
•4040 

•4245 

3859 
4061 
4265 

3879 
4081 
4286 

3899 
4101 

4307 

3919 
4122 

4327 

3939 
4142 

4348 

3959 
4163 

4369 

3979 
4183 
4390 

4000 
4204 
44" 

4O2O 
4224 
4431 

3  7  I0 
3  7  I0 
3  7  10 

13  17 
14  17 
14  17 

24 
25 
26 

•4452 
•4663 
•4877 

4473 
4684 

4899 

4494 
4706 
4921 

45i5 

4727 

4942 

4536 
4748 
4964 

4557 
477° 
4986 

4578 
479i 
5008 

4599 
4813 
5029 

4621 
4834 
5051 

4642 
4856 

5°73 

4  7  10 
4  7  ii 
4  7  ii 

14  18 
14  18 
15  18 

27 
28 
29 

•5095 
•5317 

'5543 

5"7 
5340 
5566 

5139 

5362 
5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 
5430 
5658 

5228 
5452 
5681 

5250 
5475 
5704 

5272 
5498 
5727 

5295 
5520 
575° 

4  7  ii 
4  8  ii 

4  8  12 

15  18 
15  19 
15  19 

30 

'5774 

5797 

5820 

5844 

5867 

5890 

59H 

5938 

596i 

5985 

4  8  12 

16  20 

31 
32 
33 

•6009 
•6249 
•6494 

6032 
6273 
6519 

6056 
6297 
6544 

6080 
6322 
6569 

6104 
6346 
6594 

6128 

6371 
6619 

6152 

6395 
6644 

6176 
6420 
6669 

6200 

6445 
6694 

6224 
6469 
6720 

4  8  12 

4  8  12 
4  8  13 

16  20 
16  20 

17  21 

34 
35 
36 

'6745 
•7002 
7265 

6771 
7028 
7292 

6796 
7054 
7319 

6822 
7080 
7346 

6847 
7107 

7373 

6873 
7!33 
7400 

6899 
7*59 
7427 

6924 
7186 
7454 

6950 
7212 
748i 

6976 

7239 
7508 

4  9  13 
4  9  13 
5  9  H 

17  21 
18  22 

18  23 

37 
38 
39 

7536 
7813 
•8098 

7563 
7841 
8127 

7590 
8156 

7618 
7898 
8185 

7646 
7926 
8214 

7673 
7954 
8243 

7701 
7983 
8273 

7729 
8012 
8302 

7757 
8040 

8332 

7785 
8069 
8361 

5  9  H 
5  I0  M 
5  10  15 

18  23 
19  24 
20  24 

40 

•8391 

8421 

8451 

8481 

8511 

8541 

857i 

8601 

8632 

8662 

5  1°  '5 

20  25 

41 
42 
43 

•8693 
•9004 
•9325 

8724 
9036 
9358 

8754 
9067 

9391 

8785 
9099 
9424 

8816 
9131 

9457 

8847 
9163 
9490 

8878 
9195 
9523 

8910 
9228 
9556 

8941 
9260 
9590 

8972 

9293 
9623 

5  10  16 
5  "  16 
6  ii  17 

21  26 

21  27 

22  28 

44 

•9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

6  ii  17 

23  29 

*  From  Bottomley's  Four  Figure  Mathematical  Tables,  by  courtesy  of  The  Macmillan  Company. 


TABLES 


239 


TABLE  XXI.  —  NATURAL  TANGENTS  (Concluded). 


0' 

6' 

12' 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

45° 

I  -0000 

0035 

0070 

0105 

0141 

0176 

O2I2 

0247 

0283 

0319 

6  12  18 

24  30 

46 
47 
48 

1-0355 
1-0724 
1-1106 

0392 
0761 

"45 

0428 

0799 
1184 

0464 
0837 
1224 

0501 

0875 
1263 

0538 
0913 

1303 

0575 
0951 
1343 

0612 
0990 
1383 

0649 
1028 
1423 

0686 
1067 
1463 

6  12  18 
6  13  19 
7  13  20 

25  3i 
25  32 
26  33 

49 
50 
51 

•1504 
•1918 

•2349 

1544 
1960 

2393 

1585 

2OO2 

2437 

1626 

2045 
2482 

1667 
2088 
2527 

1708 
2131 

2572 

1750 
2174 
2617 

1792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 

2753 

7  H  21 

7   14   22 

8  15  23 

28  34 
29  36 
30  38 

52 
53 
54 

•2799 
•3270 
•3764 

2846 
3319 
3814 

2892 
3367 
3865 

2938 
34i6 
3916 

2985 
3465 
3968 

3032 
35H 
4019 

3079 
3564 
4071 

3127 

3613 
4124 

3i75 
3663 
4176 

3222 

37i3 
4229 

8  16  23 
8  16  25 
9  17  26 

3i  39 
33  4i 

34  43 

55 

•4281 

4335 

4388 

4442 

4496 

4550 

4605 

4659 

4715 

4770 

9  18  27 

36  45 

56 

57 
58 

•4826 
•5399 

•6003 

4882 
5458 
6066 

4938 
5517 
6128 

4994 
5577 
6191 

5051 
5637 
6255 

5108 

5697 
6319 

5166 

mi 

5224 
5818 
6447 

5282 
5880 
6512 

5340 
594i 
6577 

10  19  29 
10  20  30 

II   21   32 

38  48 
40  50 
43  53 

59 
60 
61 

•6643 
•7321 

•8040 

6709 

739i 
8115 

6775 
7461 
8190 

6842 
7532 
8265 

6909 
7603 
8341 

6977 
7675 
8418 

7045 
7747 
8495 

7113 

7820 

8572 

7182 

7893 
8650 

7251 
7966 
8728 

ii  23  34 
12  24  36 
13  26  38 

45  56 
48  60 

51  64 

62 
63 
64 

1-8807 
1-9626 
2-0503 

8887 
9711 
0594 

8967 

9797 
0686 

9047 

9883 
0778 

9128 
9970 
0872 

9210 
0057 
0965 

9292 
0145 
1060 

9375 
0233 
"55 

9458 
0323 
1251 

9542 

041; 
1348 

14  27  41 
15  29  44 
16  31  47 

55  68 

58  73 
63  78 

65 

2-1445 

1543 

1642 

1742 

1842 

1943 

2045 

2148 

2251 

2355 

17  34  51 

68  85 

66 
67 
68 

2-2460 
2-3559 
2'475  i 

2566 
3673 
4876 

2673 

3789 
5002 

2781 
3906 
5129 

2889 
4023 

5257 

2998 
4142 
5386 

3109 
4262 

5517 

3220 
4383 
5649 

3332 
45°4 
5782 

3445 
4627 
59i6 

18  37  55 
20  40  60 
22  43  65 

74  92 
79  99 
87  108 

69 
70 
71 

2-6051 
27475 
2-9042 

6187 
7625 
9208 

6325 
7776 

9375 

6464 
7929 
9544 

6605 
8083 

97H 

6746 
8239 

9887 

6889 
8397 
0061 

7°34 
8556 
0237 

7179 
8716 
0415 

7326 
8878 

0595 

24  47  7i 
26  52  78 

29  58  87 

95  "8 
104  130 

"5  !44 

72 
73 
74 

3-0777 
3-2709 

3-4874 

0961 
2914 
5I05 

1146 
3122 
5339 

1334 
3332 
5576 

1524 
3544 
5816 

1716 

3759 
6059 

1910 

3977 
6305 

2106 
4197 
6554 

'2305 
4420 
6806 

2506 
4646 
7062 

32  64  96 
36  72  108 

41   82  122 

129  161 
144  180 
162  203 

75 

3-732I 

7583 

7848 

8118 

8391 

8667 

8947 

9232 

9520 

9812 

46  94  139 

i  86  232 

76 

77 
78 

4-0108 
4-33I5 
4-7046 

0408 
3662 

7453 

0713 
4015 
7867 

IO22 

4374 
8288 

1335 
4737 
8716 

l653 
5I07 
9152 

1976 
5483 
9594 

2303 
5864 
0045 

2635 
6252 

0504 

2972 
6646 
0970 

53  107  i  60 
62  124  186 
73  146  219 

214  267 
248  310 

292  365 

79 
80 
81 

5-I446 
5-67I3 
6-3138 

1929 

7297 
3859 

2422 
7894 
4596 

2924 
8502 
5350 

3435 
9124 
6122 

3955 
9758 
6912 

4486 
0405 
7920 

5026 

5578 

6140 
2432 
0264 

87  175  262 

350  437 

1066 
8548 

1742 
9395 

Difference-columns 
cease  to  be  useful,  owing 
to  the  rapidity  with 
which  the  value  of  the 
tangent  changes. 

82 
83 
84 

r"54 
8-1443 
9-5H4 

2066 
2636 
9-677 

3002 
3863 
9-845 

3962 
5126 

IO-O2 

4947 
6427 

10-20 

5958 
7769 
10-39 

6996 
9152 
10-58 

8062 

0579 
10-78 

9158 
2052 
10-99 

0285 
3572 

11-20 

85 

n-43 

11-66 

11-91 

12-16 

12-43 

12-71 

13-00 

13-30 

13-62 

I3-95 

86 
87 
88 

14-30 
19-08 
28-64 

14-67 

I9-74 
30-14 

15-06 
20-45 
31-82 

I5-46 
21-20 

3J69 

15-89 

22-02 
35-8o 

16-35 
22-90 
38-19 

16-83 
23-86 
40-92 

I7-34 
24-90 
44-07 

17-89 
26-03 

47-74 

18-46 
27-27 
52-08 

89 

57'29 

63-66 

71-62 

81-85 

95-49 

114-6 

143-2 

191-0 

286-5 

573-0 

240 


THE   THEORY  OF  MEASUREMENTS 


*  TABLE  XXII.  —  NATURAL  COTANGENTS. 


O' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

Difference-columns 
not  useful  here,  owing 
to  the  rapidity  with 
which  the  value  of  the 
cotangent  changes. 

0° 

Inf. 

573-o 

286-5 

191-0 

143-2 

114-6 

95'49 

81-85 

71-62 

63-66 

1 

2 
3 

57-29 
28-64 
19-08 

52-08 
27-27 
18-46 

4774 
26-03 

17-89 

44-07 
24-90 
17-34 

40-92 
23-86 
16-83 

38-19 
22-90 

i6'35 

35-80 
22-02 
15-89 

33-69 

2  1  -2O 

31-82 
20-45 
15-06 

19-74 
14-67 

4 
5 
6 

14-30 
ii'43 
9-5I44 

I3-95 

II'2O 
3572 

13-62 
10-99 
2052 

13-3° 
10-78 

0579 

13-00 
10-58 

9152 

12-71 
10-39 

7769 

12-43 

10-20 
6427 

I2'l6 
10-02 
5126 

11-91 

9-845 
3863 

u-66 
9-677 
2636 

7 
8 
9 

8-1443 
7'"54 
6-3138 

0285 
0264 
2432 

9158 

9395 
1742 

8062 
8548 
1066 

6996 
7920 
0405 

5958 
6912 
97S8 

4947 
6122 
9124 

3962 

5350 
8502 

3002 
4596 
7894 

2066 
3859 
7297 

10 

5-67I3 

6140 

5578 

5026 

4486 

3955 

3435 

2924 

2422 

1929 

123 

4  5 

11 
12 
13 

4-7046 
4-33I5 

0970 
6646 
2972 

0504 
6252 
2635 

0045 
5864 
2303 

9594 
5483 
1976 

9152 
5107 
1653 

8716 
4737 
1335 

8288 

4374 

1022 

7867 
4015 
0713 

7453 
3662 
0408 

74  148  222 

63  125  i  88 
53  107  160 

296  370 
252  314 
214  267 

14 
15 
16 

4-0108 
J4874 

9812 
7062 
4646 

9520 
6806 
4420 

9232 
6554 
4197 

8947 
6305 

3977 

8667 
6059 
3759 

5816 
3544 

8118 
5576 
3332 

7848 

5339 
3122 

7583 
5105 
29H 

46  93  139 

41   82  122 

36  72  108 

i  86  232 
163  204 
144  180 

17 
18 
19 

3-2709 

3-0777 
2-9042 

2506 

°595 
8878 

2305 

0415 
8716 

2106 
0237 
8556 

1910 
0061 
8397 

1716 

9887 
8239 

£524 
9714 
8083 

1334 
9544 
7929 

1146 

9375 
7776 

0961 
9208 
7625 

32  64  96 

29  58  87 
26  52  78 

129  161 

"5  *44 
104  130 

2*7475 

7326 

7179 

7°34 

6889 

6746 

6605 

6464 

6325 

6187 

24  47  7i 

95  "8 

21 
22 
23 

2-6051 
2-475  * 
2-3559 

5916 
4627 

3445 

5782 
45°4 
3332 

5649 
4383 
3220 

5517 
4262 
3109 

5386 
4142 
2998 

5257 
4023 
2889 

3906 
2781 

5002 

3789 
2673 

4876 
3673 
2566 

22  43  65 
20  40  60 
18  37  55 

87  108 

79  99 
74  92 

24 
25 
26 

~27~ 
28 
29 

2-2460 
2-1445 
2-0503 

2355 
1348 

0413 

2251 
1251 

0323 

2148 
"55 
0233 

2045 
1060 

0145 

1943 
0965 

0057 

1842 
0872 
9970 

1742 
0778 
9883 

1642 
0686 

9797 

1543 
0594 
97" 

17  34  5i 
16  31  47 
15  29  44 

68  85 
63  78 
58  73 

1-9626 
1-8807 
1-8040 

9542 
8728 
7966 

9458 
8650 

7893 

9375 
8572 
7820 

9292 
8495 
7747 

9210 

8418 
7675 

9128 
8341 
7603 

9047 
8265 

7532 

8967 
8190 
7461 

8887 
8115 
739i 

14  27  41 
i3  26  38 
12  24  36 

55  68 
51  64 
48  60 

30 

1-7321 

7251 

7182 

7"3 

7°45 

6977 

6909 

6842 

6775 

6709 

ii  23  34 

45  56 

31 
32 
33 

1-6643 
1-6003 
1-5399 

6577 
5340 

6512 
5880 
5282 

6447 
5818 

5224 

6383 

mi 

6319 
5697 
5108 

6255 
5637 
5051 

6191 
5577 
4994 

6128 

5517 
4938 

6066 

5458 
4882 

II   21   32 
10   20   30 

10  19  29 

43  53 
40  5° 
38  48 

34 
35 
36 

1-4826 
1-4281 
1-3764 

4770 
4229 
3713 

4715 
4176 
3663 

4659 
4124 

3613 

4605 
4071 
3564 

4550 
4019 
35H 

4496 
3968 
3465 

4442 
3916 

4388 
3865 
3367 

4335 
3814 
3319 

9  18  27 
9  17  26 
8  16  25 

36  45 
34  43 
33  4i 

37 
38 
39 

1-3270 
1-2799 
1-2349 

3222 

2753 
2305 

2708 
2261 

3^27 
2662 
2218 

3079 
2617 
2174 

3032 
2572 
2131 

2985 
2527 
2088 

2938 
2482 
2045 

2892 
2437 

2OO2 

2846 

2393 
1960 

8  16  23 
8  15  23 

7   14   22 

31  39 

30  38 
29  36 

40 

1-1918 

1875 

1833 

1792 

!75o 

1708 

1667 

1626 

1585 

1544 

7  *4  21 

28  34 

41 
42 
43 

1-1504 
1-1106 
1-0724 

1463 
1067 
0686 

1423 
1028 
0649 

1383 
0990 
0612 

1343 
0951 
0575 

1303 
0913 
0538 

1263 
0875 
0501 

1224 

0837 
0464 

1184 

0799 
0428 

"45 
0761 
0392 

7  13  20 
6  13  19 
6  12  18 

26  33 
25  32 
25  31 

44 

1-0355 

0319 

0283 

0247 

O2I2 

0176 

0141 

0105 

0070 

0035 

6  12  18 

24  30 

N.B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 
*  From  Bottomley's  Four  Figure  Mathematical  Tables,  by  courtesy  of  The  Macmillan  Company. 


TABLES 


241 


TABLE  XXII.  —  NATURAL  COTANGENTS  (Concluded). 


O' 

6' 

12 

18' 

24' 

3O' 

36' 

42' 

48' 

54' 

123 

4  5 

45° 

ro 

0-9965 

0-9930 

0-9896 

0-9861 

0-9827 

0-9793 

°'9759 

0-9725 

0-9691 

6  ii  17 

23  29 

46 
47 
48 

•9657 
•9325 
•9004 

9623 

9293 
8972 

9590 
9260 

8941 

9556 
9228 
8910 

9523 
9i95 
8878 

9490 
9163 
8847 

9457 
9131 
8816 

9424 
9099 
8785 

939i 
9067 

8754 

9358 
9036 
8724 

6  ii  17 
5  ii  10 
5  10  16 

22  28 
21  27 
21  26 

49 
50 
51 

•8693 
•8391 
•8098 

8662 
8361 
8069 

8632 
8332 
8040 

8601 
8302 
8012 

8571 
8273 
7983 

8541 
8243 

7954 

8511 
8214 
7926 

8481 
8185 
7898 

8451 
8156 
7869 

8421 
8127 
7841 

5  10  i5 

5  10  15 
5  I0  M 

20  25 
20  24 
19  24 

52 
53 
54 

•7813 
7536 
•7265 

7785 
7508 

7239 

7757 
748i 
7212 

7729 

$3 

7701 

7427 
7i59 

7673 
7400 

7133 

7646 

7373 
7107 

7618 
7346 
7080 

7590 
73i9 
7°54 

7563 
7292 
7028 

5  9  H 

5  9  H 
4  9  13 

18  23 
18  23 

18  22 

55 

•7002 

6976 

6950 

6924 

6899 

6873 

6847 

6822 

6796 

6771 

4  9  13 

I7  21 

56 
57 
58 

>6745 
•6494 
•6249 

6720 
6469 
6224 

6694 

6445 
6200 

6669 
6420 
6176 

6644 

6395 
6152 

6619 

6371 
6128 

6594 
6346 
6104 

6569 
6322 
6080 

6544 
6297 
6056 

6519 
6273 
6032 

4  8  13 
4  8  12 
4  8  12 

17  21 

16  20 
16  20 

59 
60 
61 

•6009 
'5774 
'5543 

5985 
5750 
5520 

596i 
5727 
5498 

5938 
57°4 
5475 

59H 

5681 

5452 

5890 
5658 
5430 

5867 
5^35 
5407 

5844 
5612 

5384 

5820 
5589 
5362 

5797 
5566 
5340 

4  8  12 
4  8  12 
4  8  ii 

16  20 

15  !9 

15  !9 

62 
63 
64 

•5317 
•5°95 
•4877 

5295 
5°73 
4856 

5272 
5051 
4834 

5250 
5029 

4813 

5228 
5008 
479i 

5206 
4986 
4770 

5184 
4964 
4748 

5161 

4942 
4727 

5*39 
4921 
4706 

5"7 
4899 
4684 

4  7  ii 

4  7  ii 
4  7  ii 

15  18 
15  18 
14  18 

65 

•4663 

4642 

4621 

4599 

4578 

4557 

4536 

4515 

4494 

4473 

4  7  10 

14  18 

66 
67 
68 

"4452 
•4245 
•4040 

443i 
4224 
4020 

4411 
4204 
4000 

4390 
4183 
3979 

4369 
4163 
3959 

4348 
4142 

3939 

4327 
4122 

3919 

4307 
4101 

3899 

4286 
4081 
3879 

4265 
4061 
3859 

371° 
3  7  10 
3  7  I0 

14  17 
14  17 
13  17 

69 
70 

71 

•3839 
•3640 

'3443 

3819 
3620 

3424 

3799 
3600 

3404 

3779 
358i 
3385 

3759 
356i 
3365 

3739 
354i 
3346 

3719 
3522 
3327 

3699 
3502 

3307 

3679 
3482 
3288 

3659 
3463 
3269 

3  7  10 
3  6  10 
3  6  10 

13  17 
13  17 
13  16 

72 
73 
74 

•3249 
•3057 
•2867 

3230 
3038 
2849 

3211 
3019 
2830 

3i9i 
3000 
2811 

3172 
2981 
2792 

3153 

2962 

2773 

3134 
2943 
2754 

3"5 
2924 
2736 

3096 
2905 
2717 

2698 

3  6  10 

369 
369 

13  16 
13  16 
13  16 

75 

•2679 

2661 

2642 

2623 

2605 

2586 

2568 

2549 

2530 

2512 

369 

12  16 

76 
77 
78 

•2493 
•2309 
•2126 

2475 
2290 
2107 

2456 
2272 
2089 

2438 

2254 
2071 

2419 
2235 
2053 

2401 
2217 
2035 

2382 

2199 
2016 

2364 
2180 
1998 

2345 
2162 
1980 

2327 
2144 
1962 

369 
369 
369 

12  15 
12  15 

12  I5 

79 
80 
81 

•1944 
•1763 
•1584 

1926 

'745 
1566 

1908 
1727 
1548 

1890 
1709 
1530 

1871 
1691 
1512 

1853 
1673 
H95 

1835 
l655 
H77 

1817 
1638 
H59 

1799 
1620 
1441 

1781 
1602 
1423 

369 
369 
369 

12  I5 
12  I5 
12  I5 

82 
83 
84 

•1405 
•1228 
•1051 

1388 

I2IO 
1033 

1370 
1192 
1016 

1352 

"75 
0998 

1334 
"57 
0981 

1317 
"39 
0963 

1299 

1122 
0945 

1281 
1104 
0928 

1263 
1086 
0910 

1246 
1069 
0892 

369 
369 
369 

12  15 
12  15 
12  15 

85 

•0875 

0857 

0840 

0822 

0805 

0787 

0769 

0752 

0734 

0717 

369 

12  I5 

86 
87 
88 

•0699 
•0524 
•0349 

0682 
0507 
0332 

0664 
0489 
03i4 

0647 
0472 
0297 

0629 

0454 
0279 

0612 

0437 
0262 

0594 
0419 
0244 

0577 
0402 
0227 

0559 
0384 
0209 

0542 
0367 
0192 

369 

369 
369 

12  15 
12  15 
12  I5 

89 

•oi75 

0157 

0140 

0122 

0105 

0087 

OC>7O 

0052 

0035 

0017 

369 

12  14 

N.B.  —  Numbers  in  difference-columns  to  be  subtracted,  not  added. 


242 


THE  THEORY  OF  MEASUREMENTS 


TABLE  XXIII.  —  RADIAN  MEASURE. 


0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

123 

4   5 

0° 

0.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

369 

12  15 

1 

0.0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

369 

12  15 

2 

0.0349 

0367 

0384 

0401 

0419 

0436 

0454 

0471 

0489 

0506 

369 

12  15 

3 

0.0524 

0541 

0559 

0576 

0593 

0611 

0628 

0646 

0663 

0681 

369 

12  15 

4 

0.0698 

0716 

0733 

0750 

0768J  0785 

0803 

0820 

0838 

0855 

369 

12  15 

5 

0.0873 

0890 

0908 

0925 

0942 

0960 

0977 

0995 

1012 

1030 

369 

12  15 

6 

0.1047 

1065 

1082 

1100 

1117 

1134 

1152 

1169 

1187 

1204 

369 

12  15 

7 

0.1222 

1239 

1257 

1274 

1292 

1309 

1326 

1344 

1361 

1379 

369 

12  15 

8 

0.1396 

1414 

1431 

1449 

1466 

1484 

1501 

1518 

1536 

1553 

369 

12  15 

9 

0.1571 

1588 

1606 

1623 

1641 

1658 

1676 

1693 

1710 

1728 

369 

12  15 

10 

0.1745 

1763 

1780 

1798 

1815 

1833 

1850 

1868 

1885 

1902 

369 

12  15 

11 

0.1920 

1937 

1955 

1972 

1990 

2007 

2025 

2042 

2059 

2077 

369 

12  15 

12 

0.2094 

2112 

2129J2147 

2164 

2182 

2199 

2217 

2234 

2251 

369 

12  15 

13 

0.2269 

2286 

230412321 

2339 

2356 

2374 

2391 

2409 

2426 

369 

12  15 

14 

0.2443 

2461 

2478  2496 

2513 

2531 

2548 

2566 

2583 

2601 

369 

12  15 

15 

0.2618 

2635 

2653 

2670 

2688 

2705 

2723 

2740 

2758 

2775 

369 

12  15 

16 

0.2793 

2810 

2827 

2845 

2862 

2880 

2897 

2915 

2932 

2950 

369 

12  15 

17 

0.2967 

2985 

3002 

3019 

3037 

3054 

3072 

3089 

3107 

3124 

369 

12  15 

18 

0.3142 

3159 

3176 

3194 

3211 

3229 

3246 

3264 

3281 

3299 

369 

12  15 

19 

0.3316 

3334 

3351 

3368 

3386 

3403 

3421 

3438 

3456 

3473 

369 

12  15 

20 

0.3491 

3508 

3526 

3543 

3560 

3578 

3595 

3613 

3630 

3648 

369 

12  15 

21 

0.3665 

3683 

3700 

3718 

3735 

3752 

3770 

3787 

3805 

3822 

369 

12  15 

22 

0.3840 

3857 

3875 

3892 

3910 

3927 

3944 

3962 

3979 

3997 

369 

12  15 

23 

0.4014 

4032 

4049 

4067 

4084 

4102 

4119 

4136 

4154 

4171 

369 

12  15 

24 

0.4189 

4206 

4224 

4241 

4259 

4276 

4294 

4311 

4328 

4346 

369 

12  15 

25 

0.4363 

4381 

4398 

4416 

4433 

4451 

4468 

4485 

4503 

4520 

369 

12  15 

26 

0.4538 

4555 

4573 

4590 

4608 

4625 

4643 

4660 

4677 

4695 

369 

12  15 

27 

0.4712 

4730 

4747 

4765 

4782 

4800 

4817 

4835 

4852 

4869 

369 

12  15 

28 

0.4887 

4904 

4922 

4939 

4957 

4974 

4992 

5009 

5027 

5044 

369 

12  15 

29 

0.5061 

5079 

5096 

5114 

5131 

5149 

5166 

5184 

5201 

5219 

369 

12  15 

30 

0.5236 

5253 

5271 

5288 

5306 

5323 

5341 

5358 

5376 

5393 

369 

12  15 

31 

0.5411 

5428 

5445 

5463 

5480 

5498 

5515 

5533 

5550 

5568 

369 

12  15 

32 

0.5585 

5603 

5620 

5637 

5655 

5672 

5690 

5707 

5725 

5742 

369 

12  15 

33 

0.5760 

5777 

5794 

5812 

5829 

5847 

5864 

5882 

5899 

5917 

369 

12  15 

34 

0.5934 

5952 

5969 

5986 

6004 

6021 

6039 

6056 

6074 

6091 

369 

12  15 

35 

0.6109 

6126 

6144 

6161 

6178 

6196 

6213 

6231 

6248 

6266 

369 

12  15 

36 

0.6283 

6301 

6318 

6336 

6353 

6370 

6388 

6405 

6423 

6440 

369 

12  15 

37 

0.6458 

6475 

6493 

6510 

6528 

6545 

6562 

6580 

6597 

6615 

369 

12  15 

38 

0.6632 

6650 

6667 

6685 

6702 

6720 

6737 

6754 

6772 

6789 

369 

12  15 

39 

0.6807 

6824 

6842 

6859 

6877 

6894 

6912 

6929 

6946 

6964 

369 

12  15 

40 

0.6981 

6999 

7016 

7034 

7051 

7069 

7086 

7103 

7121 

7138 

369 

12  15 

41 

0.7156 

7173 

7191 

7208 

7226 

7243 

7261 

7278 

7295 

7313 

369 

12  15 

42 

0.7330 

7348 

7365 

7383 

7400 

7418 

7435 

7453 

7470 

7487 

369 

12  15 

43 

0.7505 

7522 

7540 

7557 

7575 

7592 

7610 

7627 

7645 

7662 

369 

12  15 

44 

0.7679 

7697 

7714 

7732 

7749 

7767 

7784 

7802 

7819 

7837 

369 

12  15 

(Bottomley,  "  Four  Fig.  Math.  Tables.") 


TABLES 
TABLE  XXIII. — RADIAN  MEASURE  (Concluded). 


243 


0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1   2   3 

4    5 

45° 

0.7854 

7871 

7889 

7906 

7924 

7941 

7959 

7976 

7994 

8011 

369 

12  15 

46 

0.8029 

8046 

8063 

8081 

8098 

8116 

8133 

8151 

8168 

8186 

369 

12  15 

47 

0.8203 

8221 

8238 

8255 

8273 

8290 

8308 

8325 

8343 

8360 

369 

12  15 

48 

0.8378 

8395 

8412 

8430 

8447 

8465 

8482 

8500 

8517 

8535 

369 

12  15 

49 

0.8552 

8570 

8587 

8604 

8622 

8639 

8657 

8674 

8692 

8709 

369 

12  15 

50 

0.8727 

8744 

8762 

8779 

8796 

8814 

8831 

8849 

8866 

8884 

369 

12  15 

51 

0.8901 

8919 

8936 

8954 

8971 

8988 

9006 

9023 

9041 

9058 

369 

12  15 

52 

0.9076 

9093 

9111 

9128 

9146 

9163 

9180 

9198 

9215 

9233 

369 

12  15 

53 

0.9250 

9268 

9285 

9303 

9320 

9338 

9355 

9372 

9390 

9407 

369 

12  15 

54 

0.9425 

9442 

9460 

9477 

9495 

9512 

9529 

9547 

9564 

9582 

369 

12  15 

55 

0.9599 

9617 

9634 

9652 

9669 

9687 

9704 

9721 

9739 

9756 

369 

12  15 

56 

0.9774 

9791 

9809 

9826 

9844 

9861 

9879 

9896 

9913 

9931 

369 

12  15 

57 

0.9948 

9966 

9983 

0001 

0018 

0036 

0053 

0071 

0088 

0105 

369 

12  15 

58 

1.0123 

0140 

0158 

0175 

0193 

0210 

0228 

0245 

0263 

0280 

369 

12  15 

59 

1.0297 

0315 

0332 

0350 

0367 

0385 

0402 

0420 

0437 

0455 

369 

12  15 

60 

1.0472 

0489 

0507 

0524 

0542 

0559 

0577 

0594 

0612 

0629 

369 

12  15 

61 

1.0647 

0664 

0681 

0699 

0716 

0734 

0751 

0769 

0786 

0804 

369 

12  15 

62 

1.0821 

0838 

0856 

0873 

0891 

0908 

0926 

0943 

0961 

0978 

369 

12  15 

63 

1.0996 

1013 

1030 

1048 

1065 

1083 

1100 

1118 

1135 

1153 

369 

12  15 

64 

1.1170 

1188 

1205 

1222 

1240 

1257 

1275 

1292 

1310 

1327 

369 

12  15 

65 

1.1345 

1362 

1380 

1397 

1414 

1432 

1449 

1467 

1484 

1502 

369 

12  15 

66 

1.1519 

1537 

1554 

1572 

1589 

1606 

1624 

1641 

1659 

1676 

369 

12  15 

67 

1.1694 

1711 

1729 

1746 

1764 

1781 

1798 

1816 

1833 

1851 

369 

12  15 

68 

1.1868 

1886 

1903 

1921 

1938 

1956 

1973 

1990 

2008 

2025 

369 

12  15 

69 

1.2043 

2060 

2078 

2095 

2113 

2130 

2147 

2165 

2182 

2200 

369. 

12  15 

70 

1.2217 

2235 

2252 

2270 

2287 

2305 

2322 

2339 

2357 

2374 

369 

12  15 

71 

1.2392 

2409 

2427 

2444 

2462 

2479 

2497 

2514 

2531 

2549 

369 

12  15 

72 

1.2566 

2584 

2601 

2619 

2636 

2654 

2671 

2689 

2706 

2723 

369 

12  15 

73 

1.2741 

2758 

2776 

2793 

2811 

2828 

2846 

2863 

2881 

2898 

369 

12  15 

74 

1.2915 

2933 

2950 

2968 

2985 

3003 

3020 

3038 

3055 

3073 

369 

12  15 

75 

1.3090 

3107 

3125 

3142 

3160 

3177 

3195 

3212 

3230 

3247 

369 

12  15 

76 

1  .  3265 

3282 

3299 

3317 

3334 

3352 

3369 

3387 

$404 

3422 

369 

12  15 

77 

1  3439 

3456 

3474 

3491 

3509 

3526 

3544 

3561 

3579 

3596 

369 

12  15 

78 

1.3614 

3631 

3648 

3666 

3683 

3701 

3718 

3736 

3753 

3771 

369 

12  15 

79 

1.3788 

3806 

3823 

3840 

385& 

3875 

3893 

3910 

3928 

3945 

369 

12  15 

80 

1.3963 

3980 

3998 

4015 

4032 

4050 

4067 

4085 

4102 

4120 

369 

12  15 

81 

1.4137 

4155 

4172 

4190 

4207 

4224 

4242 

4259 

4277 

4294 

369 

12  15 

82 

1.4312 

4329 

4347 

4364 

4382 

4399 

4416 

4434 

4451 

4469 

369 

12  15 

83 

1.4486 

4504 

4521 

4539 

4556 

4573 

4591 

4608 

4626 

4643 

369 

12  15 

84 

1.4661 

4678 

4696 

4713 

4731 

4748 

4765 

4783 

4800 

4818 

369 

12  15 

85 

1.4835 

4853 

4870 

4888 

4905 

4923 

4940 

4957 

4975 

4992 

369 

12  15 

86 

1.5010 

5027 

5045 

5062 

5080 

5097 

5115 

5132 

5149 

5167 

369 

12  15 

87 

1.5184 

5202 

5219 

5237 

5254 

5272 

5289 

5307 

5324 

5341 

369 

12  15 

88 

1.5359 

5376 

5394 

5411 

5429 

5446 

5464 

5481 

5499 

5516 

369 

12  15 

89 

1.5533 

5551 

5568 

5586 

5603 

5621 

5638 

5656 

5673 

5691 

369 

12  15 

INDEX. 


A. 

Absolute  measurements,  5. 
Accidental  errors,  axioms  of,  29. 

errors,  criteria  of,  121. 

errors,  definition  of,  26 

errors,  law  of,  29,  35. 
Adjusted  effects,  149. 
Adjustment  of  the  angles  about  a 
point,  81. 

of  the  angles  of  a  plane  triangle,  93. 

of  instruments,  15,  183. 

of  measurements,  21,  42,  63,  72. 
Applications  of  the  method  of  least 

squares,  203. 

Arithmetical     mean,      characteristic 
errors  of,  51. 

mean,  principle  of,  29. 

mean,  properties  of,  42. 
Average  error,  defined,  44. 
Axioms  of  accidental  errors,  29. 

B. 

Best  magnitudes  for  components, 
fundamental  principles,  165. 
general  solutions,  167. 
practical  examples,  173. 
special  cases,  170. 

C. 

Characteristic  errors,  defined,  44. 

errors,  computation  of,  53,  57,  66, 
71,  99,  101,  112,  114. 

errors  of  the  arithmetical  mean,  51. 

errors,  relations  between,  49. 
Chauvenet's  criterion,  127. 
Computation  checks  for  normal  equa- 
tions, 83. 
Conditioned  measurements,  17. 

quantities,  determination  of,  92. 


Constant  errors,  elimination  of,  117. 

errors,  defined,  23. 
Conversion  factor,  defined,  3. 

factor,  determination  of,  8. 
Correction  factors,  defined,  131. 
Criteria  of  accidental  errors,  121. 
Criticism  of  published  results,  proper 

basis  for,  117. 

Curves,  use  of,  in  reducing  observa- 
tions, 198. 

D. 

Dependent  measurements,  17. 
Derived  measurements,  defined,  12. 
measurements,  precision  of,  135. 
quantities,  defined,  95. 
quantities,  errors  of,  99. 
units,  4. 

Dimensions  of  units,  5. 
Direct  measurements,  defined,  11. 
measurements,  precision  of,  130. 
Discussion    of     completed    observa- 
tions, 117. 
of  proposed  measurements,  general 

problem,  145. 

of  proposed  measurements,  prelim- 
inary considerations,  144. 
of  proposed  measurements,  primary 
condition,  146. 

E. 

Effective     sensitiveness     of     instru- 
ments, 183. 

Equal  effects,  principle  of,  147. 
Equations,  observation,  74. 

normal,  75. 
Error,  average,  44. 

fractional,  101. 

mean,  46. 

probable,  47. 


245 


246 


INDEX 


Error,  — Continued. 

unit,  31. 

weighted,  67. 
Errors,  accidental,  26. 

characteristic,  44. 

constant,  23. 

definition  of,  18. 

of  adjusted  measurements,  105. 

of  derived  quantities,  99. 

of  multiples  of  a  measured  quan- 
tity, 98. 

of  the  algebraic  sum  of  a  number 
of  terms,  95. 

of  the   product   of  a  number  of 
factors,  102. 

percentage,  104. 

personal,  25. 

propagation  of,  95. 

systematic,  118. 

systems  of,  33. 
Examples,  see  Numerical  examples. 

F. 

Fractional  error,  defined,  101. 

error  of  the  product  of  a  number 

of  factors,  102. 
Free  components,  169. 
Functional   relations,    determination 

of,  15,  195,  198,  203. 
Fundamental  units,  4. 

G. 

Gauss's  method  for  the  solution  of 

normal  equations,  84. 
General  mean,  63. 

principles,  1. 
Graphical  methods  of  reduction,  198. 

I. 

Independent  measurements,  17. 
Indirect  measurements,  11. 
Intrinsic     sensitiveness     of     instru- 
ments, 183. 


Law  of  accidental  errors,  29,  35. 

Laws  of  science,  2. 

Least  squares,  method  of,  72. 


M. 

Mathematical   constants,  use  of,  in 

computations,  153. 
Mean  error,  defined,  46. 
Measurement,  defined,  2. 
Measurements,  absolute,  5. 
adjustment  of,  21,  42,  63,  72. 
derived,  12. 
direct,  11. 

discussion  of,  117,  144. 
independent,  dependent,  and  con- 
ditioned, 17. 
indirect,  11. 

precision  of,  19,  130,  135. 
weights  of,  61. 

Method    of    least    squares,    applica- 
tions of,  203. 

of  least  squares,  fundamental  prin- 
ciples of,  72. 
Mistakes,  26. 

N. 

Negligible  components,  154. 

effects,  151. 

Normal       equations,       computation 
checks  for,  83. 

equations,  derivation  of,  75. 

equations,    solution    by    determi- 
nants, 114. 

equations,     solution     by     Gauss's 
method,  84. 

equations,  solutions  by  indetermi- 
nate multipliers,  105. 

equations,    solution   with   two   in- 
dependent variables,  78. 
Numeric,  defined,  2. 
Numerical  examples: 

Adjustment  of  angles  about  a  point, 
81. 

Adjustment  of  angles  of  a  plane 
triangle,  93. 

Application  of   Chauvenet's  crite- 
rion, 129. 

Best  magnitudes  for  components, 
173,  175,  180. 

Characteristic     errors     of     direct 
measurements,  56,  70. 


INDEX 


247 


Numerical  examples  —  Continued. 

Coefficient  of  linear  expansion,  78. 

Discussion  of  proposed  measure- 
ment, 157. 

Effective  sensitiveness  of  potenti- 
ometer, 190. 

Errors  of  a  derived  quantity,  101. 

Fractional  errors,  101. 

Precision  of  completed  measure- 
ment, 140. 

Probable  errors  of  adjusted  meas- 
urements, 113,  115. 

Probable  error  of  general  mean,  69. 

Propagation  of  errors,  101. 

Solution  of  normal  equations  by 
Gauss's  method,  88. 

Weighted  direct  measurement,  69. 

O. 

Observation,  denned,  15. 

equations,  74. 

standard,  62. 
Observations,  record  of,  16. 

report  of,  211. 

representation  of,  by  curves,  198. 


P. 

Percentage  errors,  104. 
Personal  equation,  26. 

errors,  25. 

Physical  tables,  use  of,  138. 
Precision  constant,  35. 
Precision  of  derived  measurements, 

135. 

of  direct  measurements,  130. 
of  measurement,  denned,  19. 
Precision  measure,  denned,  132. 
Preliminary  considerations  for  select- 
ing methods  of  measurement, 
144. 

Primary  condition,  146. 
Principle  of  the  arithmetical  mean, 

29. 

of  equal  effects,  147. 
Probability  curve,  32. 
function,  34. 


Probability  curve  —  Continued. 
function,  comparison  with  experi- 
ence, 40. 
integral,  37. 
of  large  residuals,  124. 
of  residuals,  30. 
principles  of,  28. 
Probable  error,  denned,  47. 

error    of    adjusted  measurements, 

111,  112,  116. 

error  of  the  arithmetical  mean,  53. 
error  of  direct  measurements,  com- 
putation of,  54,  55,  57. 
error  of  the  general  mean,  66,  68. 
error  of  a  single  observation,  54, 

68,  108. 

error  of  a  standard  observation,  62. 
Propagation  of  errors,  95. 
Publication,  209. 

R. 

Research,     fundamental     principles, 
192. 

general  methods,  193. 
Residuals,  defined,  27. 

distribution  of,  29. 

probability  of,  30,  124. 

S. 

Sensitiveness  of  methods  and  instru- 
ments, 183. 

Separate  effects  of  errors,  133,  135. 
Setting  of  instruments,  15. 
Sign-changes,  defined,  123. 
Sign-follows,  defined,  123. 
Significant  figures,  use  of,  19,  58. 
Slugg,  defined,  9. 

Special  functions,  treatment  of,  155. 
Standard  observation,  defined,  62. 
Systematic  errors,  defined,  118. 
Systems  of  errors,  33. 
of  units,  7. 

T. 

Tables,  list  of,  ix. 
Transformation  of  units,  8. 
Treatment  of  special  functions,  155. 


248  INDEX 

U.  W. 

Unit  error,  31.  Weighted  errors,  67. 

Units,  c.g.s.  system,  7.  mean,  63. 

dimensions  of,  5.  Weights  of  adjusted  measurements, 

engineer's  system,  7.  105,  112,  114. 

fundamental  and  derived,  4.  of  direct  measurements,  61. 

systems  in  general  use,  7. 

transformation  of,  8. 
Use  of  physical  tables,  138. 

significant  figures,  19,  58. 


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